1 Business Understanding

Customer churn is a term that refers to the event that an existing customer (or client) of a company decides to separate from the company. This could mean that the customer switches to buying from a competitor, or simply, that the customer stops doing business with the company.

1.1 Modeling customer churn

It is important for digitally enabled companies to track customer churn. To model this aspect of customer behavior the first approach that comes to mind is supervised learning. A dataset will contain some features of customers and a target variable: Whether a customer has churned or not. A classification model could be helpful to identify customers that might churn some or other time. But will this be useful? Eventually, all customers will churn at some future time. What might be much more useful is to predict the lifetime of a customer, in other words, how long a customer will stay engaged with the company.

When predicting the lifetime of a customer, we change from a classification to a regression model as we need to predict a numerical value (lifetime). This regression approach, however, has a serious drawback. Most customers will not have churned, at a specific point in time. This means for all these datapoints we do not have a target value. We can choose to eliminate these datapoints from the dataset, but they represent the larger part of it! In addition, the fact that all these customers are still engaged with the company represent valuable information. It will be beneficial to have a method that can handle this complication. Such a method is that of Survival Analysis.

Survival analysis focuses on a unique kind of outcome variable: the time until an event occurs. In our present situation this event is the 'death' or churn of a customer. This will allow us to predict the time until a specific customer churns. For all customers in the dataset who are still active, we simply do not know when they will churn, and we simply flag their time-to-churn as censored. A customer record may also be censored in other cases, e.g. when a customer's data are lost, or when a customer engaged with the company earlier than the date range of the dataset.

1.2 Survival and Censoring Times

We assume that for each customer, there is a

  • Survival time, $T$

This is the time at which the event of interest occurs. Survival time is also referred to as event time or failure time. In our present case this refers to the duration from engagement with the company until the cancellation of subscription, say (i.e. churn.

  • Censoring time $C$

This is time at which censoring occurs. This could be the time at which the last assessment of the cusomer is undertaken, or the time at which the dataset is compiled (given the customer is still engaged).

In our dataset, we will have two outcome variables:

  • Duration $Y$

We observe a duration which could be either a survival time $T$, or else a censoring time $C$. In fact, we observe the random variable $Y = min(T, C)$. If $T<C$ we observe the survival time $T$. Alternatively, if $T>C$ we observe the censoring time $C$.

  • Event/Status indicator $\delta$

In addition to observing a duration, we also observe a status or event indicator,

$$ \delta = \begin{cases} 1 \text{ if } T \leq C\\ 0 \text{ if } T > C \end{cases} $$

When we observe a survival time $\delta=1$, and if we instead observe a censoring time $\delta=0$.

  • Dataset

The dataset consists of a set of datapoints $(x_1, y_1, \delta_1), ..., (x_i, y_i, \delta_i), ..., (x_n, y_n, \delta_n)$ where

  • $x_i$ is the feature vector
  • $y_i$ is the duration, either a survival time or a censoring time
  • $\delta_i$ is the status/event indicator

This means we actually have two outcome variables: $y$ and $\delta$. The $x$'s are the usual predictor points.

1.3 Other survival analysis terminology

  • Usual statistical functions

The survival time $T$ is a random variable and the usual statistical properties apply. First there is the cumulative distribution function (cdf) of $T$:

$F(t) = Pr\{T \leq t\}$

Next, there is the probability density function (pdf) of $T$:

$f(t)$

which is also known as the event density function, i.e. the rate of events per unit time.

The usual relationships exist between $F(t)$ and $f(t)$:

$F(t) = \int_{-\infty}^{t}f(u)du$

$f(t) = \frac{dF(t)}{dt}$

  • Survival function/probability

$S(t) = Pr\{T > t\}$

Survival probability in $t$ is the probability that the survival time, $T$ exceeds $t$, i.e. the probability to survive beyond $t$.

Note that

$1 - F(t) = S(t)$, and

$f(t) = -\frac{dS(t)}{dt}$

  • Survival function estimate

When we estimate the survival probability with numerical techniques, we usually refer to the survival function estimate

$\hat{S}(t)$

  • Hazard function

The hazard function expresses the conditional probability that the event will occur within $t, t+dt$, given that it has not occurred before

$h(t) = \lim_{\Delta t \to 0} \frac{Pr\{t < T \leq t+\Delta t | T > t \}}{\Delta t} = \frac{f(t)}{S(t)} = -\frac{d}{dt}S(t)$

$h(t)$ is the event rate in the instant after time $t$, given survival past that time.

  • Cumulative hazard function

$H(t) = \int_{0}^{t}h(u)du$

  • Some relationships

$S(t) = \text{exp}[-\int_{0}^{t}h(u)du]$

$S(t) = \text{exp}[-H(t)]$

$h(t) = -[\frac{\frac{dS(t)}{dt}}{S(t)}]$

1.4 The goal of survival analysis

The goal of survival analysis may be simply stated as to:

Estimate the time to the event of interest, $T_j$ (survival time), for a new instance $j$ with feature predictors denoted by $X_j$.

2 Data Understanding

!pip install lifelines
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import warnings
warnings.filterwarnings('ignore') #6
%matplotlib inline
%config InlineBackend.figure_format = 'retina'
from matplotlib import pyplot as plt
import seaborn as sns #7
import pandas as pd
pd.set_option('display.max_columns', None)
import lifelines
print('pandas:', pd.__version__)
print('lifelines:', lifelines.__version__)
pandas: 1.3.5
lifelines: 0.27.0
df = pd.read_csv('https://raw.githubusercontent.com/'
                         'treselle-systems/customer_churn_analysis/'
                         'master/WA_Fn-UseC_-Telco-Customer-Churn.csv')
# df.head(10)
df
customerID gender SeniorCitizen Partner Dependents tenure PhoneService MultipleLines InternetService OnlineSecurity OnlineBackup DeviceProtection TechSupport StreamingTV StreamingMovies Contract PaperlessBilling PaymentMethod MonthlyCharges TotalCharges Churn
0 7590-VHVEG Female 0 Yes No 1 No No phone service DSL No Yes No No No No Month-to-month Yes Electronic check 29.85 29.85 No
1 5575-GNVDE Male 0 No No 34 Yes No DSL Yes No Yes No No No One year No Mailed check 56.95 1889.5 No
2 3668-QPYBK Male 0 No No 2 Yes No DSL Yes Yes No No No No Month-to-month Yes Mailed check 53.85 108.15 Yes
3 7795-CFOCW Male 0 No No 45 No No phone service DSL Yes No Yes Yes No No One year No Bank transfer (automatic) 42.30 1840.75 No
4 9237-HQITU Female 0 No No 2 Yes No Fiber optic No No No No No No Month-to-month Yes Electronic check 70.70 151.65 Yes
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
7038 6840-RESVB Male 0 Yes Yes 24 Yes Yes DSL Yes No Yes Yes Yes Yes One year Yes Mailed check 84.80 1990.5 No
7039 2234-XADUH Female 0 Yes Yes 72 Yes Yes Fiber optic No Yes Yes No Yes Yes One year Yes Credit card (automatic) 103.20 7362.9 No
7040 4801-JZAZL Female 0 Yes Yes 11 No No phone service DSL Yes No No No No No Month-to-month Yes Electronic check 29.60 346.45 No
7041 8361-LTMKD Male 1 Yes No 4 Yes Yes Fiber optic No No No No No No Month-to-month Yes Mailed check 74.40 306.6 Yes
7042 3186-AJIEK Male 0 No No 66 Yes No Fiber optic Yes No Yes Yes Yes Yes Two year Yes Bank transfer (automatic) 105.65 6844.5 No

7043 rows × 21 columns

df.info()
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 7043 entries, 0 to 7042
Data columns (total 21 columns):
 #   Column            Non-Null Count  Dtype  
---  ------            --------------  -----  
 0   customerID        7043 non-null   object 
 1   gender            7043 non-null   object 
 2   SeniorCitizen     7043 non-null   int64  
 3   Partner           7043 non-null   object 
 4   Dependents        7043 non-null   object 
 5   tenure            7043 non-null   int64  
 6   PhoneService      7043 non-null   object 
 7   MultipleLines     7043 non-null   object 
 8   InternetService   7043 non-null   object 
 9   OnlineSecurity    7043 non-null   object 
 10  OnlineBackup      7043 non-null   object 
 11  DeviceProtection  7043 non-null   object 
 12  TechSupport       7043 non-null   object 
 13  StreamingTV       7043 non-null   object 
 14  StreamingMovies   7043 non-null   object 
 15  Contract          7043 non-null   object 
 16  PaperlessBilling  7043 non-null   object 
 17  PaymentMethod     7043 non-null   object 
 18  MonthlyCharges    7043 non-null   float64
 19  TotalCharges      7043 non-null   object 
 20  Churn             7043 non-null   object 
dtypes: float64(1), int64(2), object(18)
memory usage: 1.1+ MB
# need to fix TotalCharges for better understanding of this feature
df[df['TotalCharges']==' ']
customerID gender SeniorCitizen Partner Dependents tenure PhoneService MultipleLines InternetService OnlineSecurity OnlineBackup DeviceProtection TechSupport StreamingTV StreamingMovies Contract PaperlessBilling PaymentMethod MonthlyCharges TotalCharges Churn
488 4472-LVYGI Female 0 Yes Yes 0 No No phone service DSL Yes No Yes Yes Yes No Two year Yes Bank transfer (automatic) 52.55 No
753 3115-CZMZD Male 0 No Yes 0 Yes No No No internet service No internet service No internet service No internet service No internet service No internet service Two year No Mailed check 20.25 No
936 5709-LVOEQ Female 0 Yes Yes 0 Yes No DSL Yes Yes Yes No Yes Yes Two year No Mailed check 80.85 No
1082 4367-NUYAO Male 0 Yes Yes 0 Yes Yes No No internet service No internet service No internet service No internet service No internet service No internet service Two year No Mailed check 25.75 No
1340 1371-DWPAZ Female 0 Yes Yes 0 No No phone service DSL Yes Yes Yes Yes Yes No Two year No Credit card (automatic) 56.05 No
3331 7644-OMVMY Male 0 Yes Yes 0 Yes No No No internet service No internet service No internet service No internet service No internet service No internet service Two year No Mailed check 19.85 No
3826 3213-VVOLG Male 0 Yes Yes 0 Yes Yes No No internet service No internet service No internet service No internet service No internet service No internet service Two year No Mailed check 25.35 No
4380 2520-SGTTA Female 0 Yes Yes 0 Yes No No No internet service No internet service No internet service No internet service No internet service No internet service Two year No Mailed check 20.00 No
5218 2923-ARZLG Male 0 Yes Yes 0 Yes No No No internet service No internet service No internet service No internet service No internet service No internet service One year Yes Mailed check 19.70 No
6670 4075-WKNIU Female 0 Yes Yes 0 Yes Yes DSL No Yes Yes Yes Yes No Two year No Mailed check 73.35 No
6754 2775-SEFEE Male 0 No Yes 0 Yes Yes DSL Yes Yes No Yes No No Two year Yes Bank transfer (automatic) 61.90 No
df['TotalCharges'] = pd.to_numeric(df['TotalCharges'], errors='coerce')
df
customerID gender SeniorCitizen Partner Dependents tenure PhoneService MultipleLines InternetService OnlineSecurity OnlineBackup DeviceProtection TechSupport StreamingTV StreamingMovies Contract PaperlessBilling PaymentMethod MonthlyCharges TotalCharges Churn
0 7590-VHVEG Female 0 Yes No 1 No No phone service DSL No Yes No No No No Month-to-month Yes Electronic check 29.85 29.85 No
1 5575-GNVDE Male 0 No No 34 Yes No DSL Yes No Yes No No No One year No Mailed check 56.95 1889.50 No
2 3668-QPYBK Male 0 No No 2 Yes No DSL Yes Yes No No No No Month-to-month Yes Mailed check 53.85 108.15 Yes
3 7795-CFOCW Male 0 No No 45 No No phone service DSL Yes No Yes Yes No No One year No Bank transfer (automatic) 42.30 1840.75 No
4 9237-HQITU Female 0 No No 2 Yes No Fiber optic No No No No No No Month-to-month Yes Electronic check 70.70 151.65 Yes
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
7038 6840-RESVB Male 0 Yes Yes 24 Yes Yes DSL Yes No Yes Yes Yes Yes One year Yes Mailed check 84.80 1990.50 No
7039 2234-XADUH Female 0 Yes Yes 72 Yes Yes Fiber optic No Yes Yes No Yes Yes One year Yes Credit card (automatic) 103.20 7362.90 No
7040 4801-JZAZL Female 0 Yes Yes 11 No No phone service DSL Yes No No No No No Month-to-month Yes Electronic check 29.60 346.45 No
7041 8361-LTMKD Male 1 Yes No 4 Yes Yes Fiber optic No No No No No No Month-to-month Yes Mailed check 74.40 306.60 Yes
7042 3186-AJIEK Male 0 No No 66 Yes No Fiber optic Yes No Yes Yes Yes Yes Two year Yes Bank transfer (automatic) 105.65 6844.50 No

7043 rows × 21 columns

df['TotalCharges'].isnull().sum()
11
df_desc = df.describe(include="all").T
df_desc
count unique top freq mean std min 25% 50% 75% max
customerID 7043 7043 7590-VHVEG 1 NaN NaN NaN NaN NaN NaN NaN
gender 7043 2 Male 3555 NaN NaN NaN NaN NaN NaN NaN
SeniorCitizen 7043.0 NaN NaN NaN 0.162147 0.368612 0.0 0.0 0.0 0.0 1.0
Partner 7043 2 No 3641 NaN NaN NaN NaN NaN NaN NaN
Dependents 7043 2 No 4933 NaN NaN NaN NaN NaN NaN NaN
tenure 7043.0 NaN NaN NaN 32.371149 24.559481 0.0 9.0 29.0 55.0 72.0
PhoneService 7043 2 Yes 6361 NaN NaN NaN NaN NaN NaN NaN
MultipleLines 7043 3 No 3390 NaN NaN NaN NaN NaN NaN NaN
InternetService 7043 3 Fiber optic 3096 NaN NaN NaN NaN NaN NaN NaN
OnlineSecurity 7043 3 No 3498 NaN NaN NaN NaN NaN NaN NaN
OnlineBackup 7043 3 No 3088 NaN NaN NaN NaN NaN NaN NaN
DeviceProtection 7043 3 No 3095 NaN NaN NaN NaN NaN NaN NaN
TechSupport 7043 3 No 3473 NaN NaN NaN NaN NaN NaN NaN
StreamingTV 7043 3 No 2810 NaN NaN NaN NaN NaN NaN NaN
StreamingMovies 7043 3 No 2785 NaN NaN NaN NaN NaN NaN NaN
Contract 7043 3 Month-to-month 3875 NaN NaN NaN NaN NaN NaN NaN
PaperlessBilling 7043 2 Yes 4171 NaN NaN NaN NaN NaN NaN NaN
PaymentMethod 7043 4 Electronic check 2365 NaN NaN NaN NaN NaN NaN NaN
MonthlyCharges 7043.0 NaN NaN NaN 64.761692 30.090047 18.25 35.5 70.35 89.85 118.75
TotalCharges 7032.0 NaN NaN NaN 2283.300441 2266.771362 18.8 401.45 1397.475 3794.7375 8684.8
Churn 7043 2 No 5174 NaN NaN NaN NaN NaN NaN NaN
df.nunique()
customerID          7043
gender                 2
SeniorCitizen          2
Partner                2
Dependents             2
tenure                73
PhoneService           2
MultipleLines          3
InternetService        3
OnlineSecurity         3
OnlineBackup           3
DeviceProtection       3
TechSupport            3
StreamingTV            3
StreamingMovies        3
Contract               3
PaperlessBilling       2
PaymentMethod          4
MonthlyCharges      1585
TotalCharges        6530
Churn                  2
dtype: int64
[print(f"{col}: {df[col].unique()}") for col in df]
customerID: ['7590-VHVEG' '5575-GNVDE' '3668-QPYBK' ... '4801-JZAZL' '8361-LTMKD'
 '3186-AJIEK']
gender: ['Female' 'Male']
SeniorCitizen: [0 1]
Partner: ['Yes' 'No']
Dependents: ['No' 'Yes']
tenure: [ 1 34  2 45  8 22 10 28 62 13 16 58 49 25 69 52 71 21 12 30 47 72 17 27
  5 46 11 70 63 43 15 60 18 66  9  3 31 50 64 56  7 42 35 48 29 65 38 68
 32 55 37 36 41  6  4 33 67 23 57 61 14 20 53 40 59 24 44 19 54 51 26  0
 39]
PhoneService: ['No' 'Yes']
MultipleLines: ['No phone service' 'No' 'Yes']
InternetService: ['DSL' 'Fiber optic' 'No']
OnlineSecurity: ['No' 'Yes' 'No internet service']
OnlineBackup: ['Yes' 'No' 'No internet service']
DeviceProtection: ['No' 'Yes' 'No internet service']
TechSupport: ['No' 'Yes' 'No internet service']
StreamingTV: ['No' 'Yes' 'No internet service']
StreamingMovies: ['No' 'Yes' 'No internet service']
Contract: ['Month-to-month' 'One year' 'Two year']
PaperlessBilling: ['Yes' 'No']
PaymentMethod: ['Electronic check' 'Mailed check' 'Bank transfer (automatic)'
 'Credit card (automatic)']
MonthlyCharges: [29.85 56.95 53.85 ... 63.1  44.2  78.7 ]
TotalCharges: [  29.85 1889.5   108.15 ...  346.45  306.6  6844.5 ]
Churn: ['No' 'Yes']
[None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None,
 None]
# check missing values
df.isnull().sum()
customerID           0
gender               0
SeniorCitizen        0
Partner              0
Dependents           0
tenure               0
PhoneService         0
MultipleLines        0
InternetService      0
OnlineSecurity       0
OnlineBackup         0
DeviceProtection     0
TechSupport          0
StreamingTV          0
StreamingMovies      0
Contract             0
PaperlessBilling     0
PaymentMethod        0
MonthlyCharges       0
TotalCharges        11
Churn                0
dtype: int64

response variables

The role of response variables is played by 'tenure' for $y$ and 'Churn' for $\delta$.

plt.style.use('fivethirtyeight') #6
plt.rcParams['figure.figsize'] = [14, 8] #6
plt.rcParams['lines.linewidth'] = 2.5 #6
df['tenure'].hist(color='skyblue');
df['Churn'].value_counts(normalize=True).plot.bar(color=['lightgreen','red']);
df['MonthlyCharges'].hist(color='skyblue');
df['TotalCharges'].hist(color='skyblue');
Mth = sns.kdeplot(df['MonthlyCharges'][(df["Churn"]=='No')],
                color="Red", shade = True)
Mth = sns.kdeplot(df['MonthlyCharges'][(df["Churn"]=='Yes')],
                ax=Mth, color="Blue", shade= True)
Mth.legend(["No Churn", "Churn"], loc='upper right')
Mth.set_ylabel('Density')
Mth.set_xlabel('Monthly Charges')
Mth.set_title('Monthly charges by churn');
mypal = [
    'lightblue',    
    'lightpink',
    'lightcyan',
    'lightgreen',
    'khaki',
    'gold',
    'wheat',
    'peachpuff',
    'mistyrose',
    'sandybrown',
    'violet',
]
pd.crosstab(df['gender'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['SeniorCitizen'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['Partner'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['Dependents'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['Contract'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['PaperlessBilling'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['PaymentMethod'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['PhoneService'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['MultipleLines'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['InternetService'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['OnlineSecurity'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['OnlineBackup'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['DeviceProtection'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['TechSupport'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['StreamingTV'], df['Churn']).plot.bar(color=mypal, stacked=True);
pd.crosstab(df['StreamingMovies'], df['Churn']).plot.bar(color=mypal, stacked=True);
import numpy as np
# sns.heatmap(df.corr(), cmap="Paired")
# Generate a mask for the upper triangle #7
corr = df.corr()
mask = np.triu(np.ones_like(corr, dtype=np.bool))
# Generate a custom diverging colormap
cmap = sns.diverging_palette(220, 10, as_cmap=True)
# Plot correlation matrix
sns.heatmap(corr, cmap=cmap, linewidths=.5, mask=mask);
# inspect life times of a sample of customers
df_life = df[['tenure','Churn']]
df_life = df_life.replace({'Churn': {'No': False, 'Yes': True}})
df_life
tenure Churn
0 1 False
1 34 False
2 2 True
3 45 False
4 2 True
... ... ...
7038 24 False
7039 72 False
7040 11 False
7041 4 True
7042 66 False

7043 rows × 2 columns

from lifelines.plotting import plot_lifetimes
n = 40
sample = df_life.sample(n, replace=False)
durations = sample['tenure']
status = sample['Churn']

plt.figure(figsize=(18, 15));
plot_lifetimes(
    durations=durations, event_observed=status,
    #entry=None,
    #left_truncated=False,
    sort_by_duration=False,
    event_observed_color='red', 
    event_censored_color='blue', 
)
plt.xlabel('Days subscribed', fontsize=16);
plt.ylabel('Customer ID', fontsize=16);
plt.title('Customer subscription lifelines (random sample)', fontsize=20)
plt.tight_layout()

3 Data Preparation

df['customerID'].duplicated().any()
False
df = df.set_index('customerID')
df
gender SeniorCitizen Partner Dependents tenure PhoneService MultipleLines InternetService OnlineSecurity OnlineBackup DeviceProtection TechSupport StreamingTV StreamingMovies Contract PaperlessBilling PaymentMethod MonthlyCharges TotalCharges Churn
customerID
7590-VHVEG Female 0 Yes No 1 No No phone service DSL No Yes No No No No Month-to-month Yes Electronic check 29.85 29.85 No
5575-GNVDE Male 0 No No 34 Yes No DSL Yes No Yes No No No One year No Mailed check 56.95 1889.50 No
3668-QPYBK Male 0 No No 2 Yes No DSL Yes Yes No No No No Month-to-month Yes Mailed check 53.85 108.15 Yes
7795-CFOCW Male 0 No No 45 No No phone service DSL Yes No Yes Yes No No One year No Bank transfer (automatic) 42.30 1840.75 No
9237-HQITU Female 0 No No 2 Yes No Fiber optic No No No No No No Month-to-month Yes Electronic check 70.70 151.65 Yes
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
6840-RESVB Male 0 Yes Yes 24 Yes Yes DSL Yes No Yes Yes Yes Yes One year Yes Mailed check 84.80 1990.50 No
2234-XADUH Female 0 Yes Yes 72 Yes Yes Fiber optic No Yes Yes No Yes Yes One year Yes Credit card (automatic) 103.20 7362.90 No
4801-JZAZL Female 0 Yes Yes 11 No No phone service DSL Yes No No No No No Month-to-month Yes Electronic check 29.60 346.45 No
8361-LTMKD Male 1 Yes No 4 Yes Yes Fiber optic No No No No No No Month-to-month Yes Mailed check 74.40 306.60 Yes
3186-AJIEK Male 0 No No 66 Yes No Fiber optic Yes No Yes Yes Yes Yes Two year Yes Bank transfer (automatic) 105.65 6844.50 No

7043 rows × 20 columns

3.1 Event/Censor times/durations

df['tenure'] = df['tenure'].astype(float)

3.2 Status indicator

df['Churn'] = (df['Churn']=='Yes')

3.3 Numericals

df[df['TotalCharges'].isnull()]
gender SeniorCitizen Partner Dependents tenure PhoneService MultipleLines InternetService OnlineSecurity OnlineBackup DeviceProtection TechSupport StreamingTV StreamingMovies Contract PaperlessBilling PaymentMethod MonthlyCharges TotalCharges Churn
customerID
4472-LVYGI Female 0 Yes Yes 0.0 No No phone service DSL Yes No Yes Yes Yes No Two year Yes Bank transfer (automatic) 52.55 NaN False
3115-CZMZD Male 0 No Yes 0.0 Yes No No No internet service No internet service No internet service No internet service No internet service No internet service Two year No Mailed check 20.25 NaN False
5709-LVOEQ Female 0 Yes Yes 0.0 Yes No DSL Yes Yes Yes No Yes Yes Two year No Mailed check 80.85 NaN False
4367-NUYAO Male 0 Yes Yes 0.0 Yes Yes No No internet service No internet service No internet service No internet service No internet service No internet service Two year No Mailed check 25.75 NaN False
1371-DWPAZ Female 0 Yes Yes 0.0 No No phone service DSL Yes Yes Yes Yes Yes No Two year No Credit card (automatic) 56.05 NaN False
7644-OMVMY Male 0 Yes Yes 0.0 Yes No No No internet service No internet service No internet service No internet service No internet service No internet service Two year No Mailed check 19.85 NaN False
3213-VVOLG Male 0 Yes Yes 0.0 Yes Yes No No internet service No internet service No internet service No internet service No internet service No internet service Two year No Mailed check 25.35 NaN False
2520-SGTTA Female 0 Yes Yes 0.0 Yes No No No internet service No internet service No internet service No internet service No internet service No internet service Two year No Mailed check 20.00 NaN False
2923-ARZLG Male 0 Yes Yes 0.0 Yes No No No internet service No internet service No internet service No internet service No internet service No internet service One year Yes Mailed check 19.70 NaN False
4075-WKNIU Female 0 Yes Yes 0.0 Yes Yes DSL No Yes Yes Yes Yes No Two year No Mailed check 73.35 NaN False
2775-SEFEE Male 0 No Yes 0.0 Yes Yes DSL Yes Yes No Yes No No Two year Yes Bank transfer (automatic) 61.90 NaN False
df['TotalCharges'] = df['TotalCharges'].fillna(0.0)
df[df['TotalCharges'].isnull()]
gender SeniorCitizen Partner Dependents tenure PhoneService MultipleLines InternetService OnlineSecurity OnlineBackup DeviceProtection TechSupport StreamingTV StreamingMovies Contract PaperlessBilling PaymentMethod MonthlyCharges TotalCharges Churn
customerID

3.4 Categoricals

# Categoricals will be prepared later

4 Modeling

4.1 Survival Probability for the average customer

First we will use the Kaplan-Meier non-parametric model to estimate the survival probability of an average customer, i.e. not for an individual customer.

durations = df['tenure']
churns = df['Churn']
kmf = lifelines.KaplanMeierFitter()
kmf.fit(durations=durations, event_observed=churns, label='Estimate for Average Customer')
<lifelines.KaplanMeierFitter:"Estimate for Average Customer", fitted with 7043 total observations, 5174 right-censored observations>
kmf.survival_function_
Estimate for Average Customer
timeline
0.0 1.000000
1.0 0.945961
2.0 0.927835
3.0 0.913725
4.0 0.901045
... ...
68.0 0.626858
69.0 0.620135
70.0 0.609657
71.0 0.602781
72.0 0.592790

73 rows × 1 columns

fig, ax = plt.subplots(figsize=(12,7))
kmf.plot_survival_function(ax=ax)
ax.set_title('Kaplan-Meier Customer Survival Probability - All Customers')
ax.set_xlabel('tenure (Months)')
ax.set_ylabel('Survival Probability')
plt.hlines(y=0.7, xmin=0, xmax=70, linestyles='dotted', colors=['red'])
plt.show()

As seen from the chart, the survival probability $S(t)$ declines with tenure. Customer retention efforts can be costly and a company will lose profitability if such measures are applied without discretion. Let's say we want to have our customer retention procedure only kick in when $S(t)$ falls below 70%. From the chart it appears that this will have to happen when the tenure exceeds 51 months. This model represents the average customer which means customer retention will always happen when a customer's tenure exceeds 51 months.

sum(durations==0)
11
from lifelines.plotting import qq_plot
fig, axes = plt.subplots(2, 2, figsize=(10, 10))
timeline = np.linspace(0, 0.25, 100)
wf = lifelines.WeibullFitter().fit(durations+0.001, churns, label="Weibull", timeline=timeline)
lnf = lifelines.LogNormalFitter().fit(durations+0.001, churns, label="Log Normal", timeline=timeline)
# plot what we just fit, along with the KMF estimate
kmf.plot_cumulative_density(ax=axes[0][0], ci_show=False)
wf.plot_cumulative_density(ax=axes[0][0], ci_show=False)
qq_plot(wf, ax=axes[0][1])
kmf.plot_cumulative_density(ax=axes[1][0], ci_show=False)
lnf.plot_cumulative_density(ax=axes[1][0], ci_show=False)
qq_plot(lnf, ax=axes[1][1]);
from lifelines.utils import find_best_parametric_model
best_model, best_aic_ = find_best_parametric_model(durations+0.001, churns)
best_model.plot_hazard();

4.2 Survival Probability for the average customer based on segments

The previous approach is unfocused and may lead to a lot of waste. We can improve the effectiveness somewhat by studying the survival functions for different segments of the customer population. Segmentation can be performed based on the classes in categorical predictors. Let's demonstrate this approach by making use of the 'InternetService' predictor.

#create Kaplan-Meier curve plots segmented by categoricals
def plot_categorical_survival(dft, feature, duration='tenure', event='Churn', ax=None):
    for cat in dft[feature].unique():
        idxs = dft[feature]==cat
        kmft = lifelines.KaplanMeierFitter()
        kmft.fit(dft[idxs][duration], event_observed=dft[idxs][event], label=cat)
        kmft.plot(ax=ax, label=cat)
fig_pmt, ax_pmt = plt.subplots(figsize=(12,7))
plot_categorical_survival(dft=df, feature='PaymentMethod', ax=ax_pmt)
ax_pmt.set_title('Customer Churn by Payment Method')
ax_pmt.set_xlabel('tenure (Months)')
ax_pmt.set_ylabel('Survival Probability')
plt.hlines(y=0.7, xmin=0, xmax=70, linestyles='dotted', colors=['red'])
plt.show()
fig_contract, ax_contract = plt.subplots(figsize=(12,7))
plot_categorical_survival(dft=df, feature='Contract', ax=ax_contract)
ax_contract.set_title('Customer Churn by Contract Type')
ax_contract.set_xlabel('tenure (Months)')
ax_contract.set_ylabel('Survival Probability')
plt.hlines(y=0.7, xmin=0, xmax=70, linestyles='dotted', colors=['red'])
plt.show()
fig_dep, ax_dep = plt.subplots(figsize=(12,7))
plot_categorical_survival(dft=df, feature='Dependents', ax=ax_dep)
ax_dep.set_title('Customer Churn, Dependents vs. No Dependents')
ax_dep.set_xlabel('tenure (Months)')
ax_dep.set_ylabel('Survival Probability')
plt.hlines(y=0.7, xmin=0, xmax=70, linestyles='dotted', colors=['red'])
plt.show()
fig_dep, ax_dep = plt.subplots(figsize=(12,7))
plot_categorical_survival(dft=df, feature='InternetService', ax=ax_dep)
ax_dep.set_title('Customer Churn by InternetService')
ax_dep.set_xlabel('tenure (Months)')
ax_dep.set_ylabel('Survival Probability')
plt.hlines(y=0.7, xmin=0, xmax=70, linestyles='dotted', colors=['red'])
plt.show()

From the chart we see that we can be more effective by only activating the retention procedure for fiber optic customers (when their tenure exceeds about 26 months).

4.3 Estimate Hazard

Before we can use another approach to model the average customer, we need to make use of the concept of hazard. The hazard function or hazard rate was defined above. It is the event rate in the instant after $t$, given survival past $t$. In the context of customer churn the hazard is an important indicator and needs to be as low as possible. If it goes up it indicates something has changed between the company and a customer. Such a situation calls for intervention.

On a more technical level, why should we care about the hazard function? Firstly, it is related to the survival function $S(t)$. Secondly, an important technique for modeling survival time as a function of covariates (in this case customer features) depends on the hazard function. This kind of technique will be necessary to analyze an individual customer, rather than the whole population.

4.4 Nelson-Aalen estimator

The Nelson-Aalen estimator estimates the cumulative hazard rate. Like the Kaplan-Meier estimator, it is non-parametric.

4.4.1 Cumulative hazard function

from lifelines import NelsonAalenFitter
naf = NelsonAalenFitter(alpha=0.05, nelson_aalen_smoothing=False)
naf.fit(durations=durations, event_observed=churns)
naf.cumulative_hazard_.head()
NA_estimate
timeline
0.0 0.000000
1.0 0.054039
2.0 0.073201
3.0 0.088408
4.0 0.102286
naf.plot();
fig, ax = plt.subplots(figsize=(12,7))
var = 'InternetService'
for name, dft in df.groupby(var):
    idxs = df[var]==name
    naf = lifelines.NelsonAalenFitter().fit(dft['tenure'][idxs], dft['Churn'][idxs], alpha=0.05, label=name)
    naf.plot(ax=ax)
ax.set_title(var)
ax.set_xlabel('tenure (Months)')
ax.set_ylabel('cumulative hazard rate')
plt.show()

4.4.2 Hazard function

fig, ax = plt.subplots(figsize=(12,7))
var = 'InternetService'
for name, dft in df.groupby(var):
    idxs = df[var]==name
    naf = lifelines.NelsonAalenFitter().fit(dft['tenure'][idxs], dft['Churn'][idxs], alpha=0.05, label=name)
    naf.plot_hazard(ax=ax, bandwidth=6)
ax.set_title(var)
ax.set_xlabel('tenure (Months)')
ax.set_ylabel('hazard rate')
plt.show()

4.2 Survival Regression Model

So far, we have only looked at analyzing groups of customers. Now we are going to setup a model for an individual customer. This will increase the effectiveness of the customer retention mechanism. We next make use of the Cox Proportional Hazards model. This will allow us to model the survival time as a function of the covariates (i.e. the customer features).

4.2.1 Proportional Hazards Assumption

The proportional hazards assumption states

$h(t|x_i) = h_0(t) \text{exp} (\sum_{j=1}^{p}x_{ij}\beta_j)$

where $h_0(t)$, an unspecified function, is known as the baseline hazard. It is the hazard function for an individual with features $x_{i1} = ... = x_{ip} = 0$. The term proportional hazards derives from the fact that the hazard function for an individual with feature vector $x_i$ is some unknown function $h_0(t)$ times the factor $\text{exp} (\sum_{j=1}^{p}x_{ij}\beta_j)$. The latter quantity is called the relative risk for the feature vector $x_i = (x_{i1}, ..., x_{ip})^T$, relative to that for the feature vector $x_i = (0, ..., 0)^T$.

The baseline hazard function $h_0(t)$ can take on any form. The only assumption is that a one-unit increase in $x_{ij}$ corresponds to an increase in $h(t|x_i)$ by a factor of exp($\beta_j$).

4.2.2 Cox Proportional Hazards Regression

We need to do some pre-processing before applying this model.

df
gender SeniorCitizen Partner Dependents tenure PhoneService MultipleLines InternetService OnlineSecurity OnlineBackup DeviceProtection TechSupport StreamingTV StreamingMovies Contract PaperlessBilling PaymentMethod MonthlyCharges TotalCharges Churn
customerID
7590-VHVEG Female 0 Yes No 1.0 No No phone service DSL No Yes No No No No Month-to-month Yes Electronic check 29.85 29.85 False
5575-GNVDE Male 0 No No 34.0 Yes No DSL Yes No Yes No No No One year No Mailed check 56.95 1889.50 False
3668-QPYBK Male 0 No No 2.0 Yes No DSL Yes Yes No No No No Month-to-month Yes Mailed check 53.85 108.15 True
7795-CFOCW Male 0 No No 45.0 No No phone service DSL Yes No Yes Yes No No One year No Bank transfer (automatic) 42.30 1840.75 False
9237-HQITU Female 0 No No 2.0 Yes No Fiber optic No No No No No No Month-to-month Yes Electronic check 70.70 151.65 True
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
6840-RESVB Male 0 Yes Yes 24.0 Yes Yes DSL Yes No Yes Yes Yes Yes One year Yes Mailed check 84.80 1990.50 False
2234-XADUH Female 0 Yes Yes 72.0 Yes Yes Fiber optic No Yes Yes No Yes Yes One year Yes Credit card (automatic) 103.20 7362.90 False
4801-JZAZL Female 0 Yes Yes 11.0 No No phone service DSL Yes No No No No No Month-to-month Yes Electronic check 29.60 346.45 False
8361-LTMKD Male 1 Yes No 4.0 Yes Yes Fiber optic No No No No No No Month-to-month Yes Mailed check 74.40 306.60 True
3186-AJIEK Male 0 No No 66.0 Yes No Fiber optic Yes No Yes Yes Yes Yes Two year Yes Bank transfer (automatic) 105.65 6844.50 False

7043 rows × 20 columns

X = pd.get_dummies(df.drop(['Churn', 'tenure'], axis=1), drop_first=True)
y = df.loc[:, ['tenure', 'Churn']]
df_clean = pd.concat([X, y], axis=1)
df_clean
SeniorCitizen MonthlyCharges TotalCharges gender_Male Partner_Yes Dependents_Yes PhoneService_Yes MultipleLines_No phone service MultipleLines_Yes InternetService_Fiber optic InternetService_No OnlineSecurity_No internet service OnlineSecurity_Yes OnlineBackup_No internet service OnlineBackup_Yes DeviceProtection_No internet service DeviceProtection_Yes TechSupport_No internet service TechSupport_Yes StreamingTV_No internet service StreamingTV_Yes StreamingMovies_No internet service StreamingMovies_Yes Contract_One year Contract_Two year PaperlessBilling_Yes PaymentMethod_Credit card (automatic) PaymentMethod_Electronic check PaymentMethod_Mailed check tenure Churn
customerID
7590-VHVEG 0 29.85 29.85 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1.0 False
5575-GNVDE 0 56.95 1889.50 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 34.0 False
3668-QPYBK 0 53.85 108.15 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 2.0 True
7795-CFOCW 0 42.30 1840.75 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 45.0 False
9237-HQITU 0 70.70 151.65 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 2.0 True
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
6840-RESVB 0 84.80 1990.50 1 1 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 24.0 False
2234-XADUH 0 103.20 7362.90 0 1 1 1 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 0 0 72.0 False
4801-JZAZL 0 29.60 346.45 0 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 11.0 False
8361-LTMKD 1 74.40 306.60 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 4.0 True
3186-AJIEK 0 105.65 6844.50 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 66.0 False

7043 rows × 31 columns

df_clean.columns
Index(['SeniorCitizen', 'MonthlyCharges', 'TotalCharges', 'gender_Male',
       'Partner_Yes', 'Dependents_Yes', 'PhoneService_Yes',
       'MultipleLines_No phone service', 'MultipleLines_Yes',
       'InternetService_Fiber optic', 'InternetService_No',
       'OnlineSecurity_No internet service', 'OnlineSecurity_Yes',
       'OnlineBackup_No internet service', 'OnlineBackup_Yes',
       'DeviceProtection_No internet service', 'DeviceProtection_Yes',
       'TechSupport_No internet service', 'TechSupport_Yes',
       'StreamingTV_No internet service', 'StreamingTV_Yes',
       'StreamingMovies_No internet service', 'StreamingMovies_Yes',
       'Contract_One year', 'Contract_Two year', 'PaperlessBilling_Yes',
       'PaymentMethod_Credit card (automatic)',
       'PaymentMethod_Electronic check', 'PaymentMethod_Mailed check',
       'tenure', 'Churn'],
      dtype='object')

4.2.3 Split the data

import numpy as np
num_customers = df_clean.shape[0]

num_train = int(num_customers * 0.7)
train_customers = np.random.choice(df_clean.index, size=num_train, replace=False)
df_train = df_clean.loc[train_customers]

num_valid = int((num_customers - num_train) * 0.5)
valid_customers = np.random.choice(df_clean.drop(train_customers).index, size=num_valid, replace=False)
df_valid = df_clean.loc[valid_customers]

test_customers = df.drop(train_customers).drop(valid_customers).index
df_test = df_clean.loc[test_customers]
total = df_train.shape[0] + df_valid.shape[0] + df_test.shape[0]
print('Train', df_train.shape, df_train.shape[0]/total*100)
print('Dev', df_valid.shape, df_valid.shape[0]/total*100)
print('Test', df_test.shape, df_test.shape[0]/total*100)
print('Total:', total)
assert total == df.shape[0]
Train (4930, 31) 69.99858015050404
Dev (1056, 31) 14.99361067726821
Test (1057, 31) 15.007809172227743
Total: 7043
cph = lifelines.CoxPHFitter()
cph.fit(df_train, duration_col='tenure', event_col='Churn', show_progress=False)
cph.print_summary()
---------------------------------------------------------------------------
LinAlgError                               Traceback (most recent call last)
/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in _newton_rhapson_for_efron_model(self, X, T, E, weights, entries, initial_point, step_size, precision, show_progress, max_steps)
   1523             try:
-> 1524                 inv_h_dot_g_T = spsolve(-h, g, assume_a="pos", check_finite=False)
   1525             except (ValueError, LinAlgError) as e:

/usr/local/lib/python3.7/dist-packages/scipy/linalg/basic.py in solve(a, b, sym_pos, lower, overwrite_a, overwrite_b, debug, check_finite, assume_a, transposed)
    252                            overwrite_b=overwrite_b)
--> 253         _solve_check(n, info)
    254         rcond, info = pocon(lu, anorm)

/usr/local/lib/python3.7/dist-packages/scipy/linalg/basic.py in _solve_check(n, info, lamch, rcond)
     28     elif 0 < info:
---> 29         raise LinAlgError('Matrix is singular.')
     30 

LinAlgError: Matrix is singular.

During handling of the above exception, another exception occurred:

ConvergenceError                          Traceback (most recent call last)
<ipython-input-80-a2b2a571ba39> in <module>()
      1 cph = lifelines.CoxPHFitter()
----> 2 cph.fit(df_train, duration_col='tenure', event_col='Churn', show_progress=False)
      3 cph.print_summary()

/usr/local/lib/python3.7/dist-packages/lifelines/utils/__init__.py in f(model, *args, **kwargs)
     54         def f(model, *args, **kwargs):
     55             cls.set_censoring_type(model, cls.RIGHT)
---> 56             return function(model, *args, **kwargs)
     57 
     58         return f

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in fit(self, df, duration_col, event_col, show_progress, initial_point, strata, step_size, weights_col, cluster_col, robust, batch_mode, timeline, formula, entry_col)
    302             timeline=timeline,
    303             formula=formula,
--> 304             entry_col=entry_col,
    305         )
    306         return self

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in _fit_model(self, *args, **kwargs)
    614     def _fit_model(self, *args, **kwargs):
    615         if self.baseline_estimation_method == "breslow":
--> 616             return self._fit_model_breslow(*args, **kwargs)
    617         elif self.baseline_estimation_method == "spline":
    618             return self._fit_model_spline(*args, **kwargs)

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in _fit_model_breslow(self, *args, **kwargs)
    627         )
    628         if utils.CensoringType.is_right_censoring(self):
--> 629             model.fit(*args, **kwargs)
    630             return model
    631         else:

/usr/local/lib/python3.7/dist-packages/lifelines/utils/__init__.py in f(model, *args, **kwargs)
     54         def f(model, *args, **kwargs):
     55             cls.set_censoring_type(model, cls.RIGHT)
---> 56             return function(model, *args, **kwargs)
     57 
     58         return f

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in fit(self, df, duration_col, event_col, show_progress, initial_point, strata, step_size, weights_col, cluster_col, robust, batch_mode, timeline, formula, entry_col)
   1263             initial_point=initial_point,
   1264             show_progress=show_progress,
-> 1265             step_size=step_size,
   1266         )
   1267 

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in _fit_model(self, X, T, E, weights, entries, initial_point, step_size, show_progress)
   1384     ):
   1385         beta_, ll_, hessian_ = self._newton_rhapson_for_efron_model(
-> 1386             X, T, E, weights, entries, initial_point=initial_point, step_size=step_size, show_progress=show_progress
   1387         )
   1388 

/usr/local/lib/python3.7/dist-packages/lifelines/fitters/coxph_fitter.py in _newton_rhapson_for_efron_model(self, X, T, E, weights, entries, initial_point, step_size, precision, show_progress, max_steps)
   1535                             CONVERGENCE_DOCS
   1536                         ),
-> 1537                         e,
   1538                     )
   1539                 else:

ConvergenceError: Convergence halted due to matrix inversion problems. Suspicion is high collinearity. Please see the following tips in the lifelines documentation: https://lifelines.readthedocs.io/en/latest/Examples.html#problems-with-convergence-in-the-cox-proportional-hazard-modelMatrix is singular.
# Suspicion is high collinearity. Please see the following tips in the lifelines documentation: 
# https://lifelines.readthedocs.io/en/latest/Examples.html#problems-with-convergence-in-the-cox-proportional-hazard-modelMatrix is singular.

To address this problem we will make use of the variance inflation factor (VIF) approach.

from statsmodels.stats.outliers_influence import variance_inflation_factor
def calc_vif(X):
    vif = pd.DataFrame()
    vif["variables"] = X.columns
    vif["VIF"] = [variance_inflation_factor(X.values, i) for i in range(X.shape[1])]
    return(vif)
feature_vif = calc_vif(df_train.drop(['Churn', 'tenure'], axis=1))
feature_vif.sort_values('VIF')
variables VIF
3 gender_Male 1.002632
0 SeniorCitizen 1.155752
25 PaperlessBilling_Yes 1.219196
5 Dependents_Yes 1.370769
4 Partner_Yes 1.430055
23 Contract_One year 1.537983
26 PaymentMethod_Credit card (automatic) 1.576695
28 PaymentMethod_Mailed check 1.799718
27 PaymentMethod_Electronic check 1.976341
24 Contract_Two year 2.266687
2 TotalCharges 3.986928
12 OnlineSecurity_Yes 6.299991
18 TechSupport_Yes 6.460155
14 OnlineBackup_Yes 6.771743
16 DeviceProtection_Yes 6.921651
8 MultipleLines_Yes 7.275559
20 StreamingTV_Yes 23.770145
22 StreamingMovies_Yes 24.167640
7 MultipleLines_No phone service 61.569033
9 InternetService_Fiber optic 148.414936
1 MonthlyCharges 846.284329
6 PhoneService_Yes 1743.106075
11 OnlineSecurity_No internet service inf
15 DeviceProtection_No internet service inf
17 TechSupport_No internet service inf
19 StreamingTV_No internet service inf
21 StreamingMovies_No internet service inf
10 InternetService_No inf
13 OnlineBackup_No internet service inf
features_keep = feature_vif.loc[feature_vif['VIF'] < 10, 'variables'].tolist()
features_keep.extend(['tenure', 'Churn'])
features_keep
['SeniorCitizen',
 'TotalCharges',
 'gender_Male',
 'Partner_Yes',
 'Dependents_Yes',
 'MultipleLines_Yes',
 'OnlineSecurity_Yes',
 'OnlineBackup_Yes',
 'DeviceProtection_Yes',
 'TechSupport_Yes',
 'Contract_One year',
 'Contract_Two year',
 'PaperlessBilling_Yes',
 'PaymentMethod_Credit card (automatic)',
 'PaymentMethod_Electronic check',
 'PaymentMethod_Mailed check',
 'tenure',
 'Churn']
cph = lifelines.CoxPHFitter()
cph.fit(
    df_train.loc[:, features_keep], 
    duration_col='tenure', 
    event_col='Churn', 
    show_progress=False)
cph.print_summary()
model lifelines.CoxPHFitter
duration col 'tenure'
event col 'Churn'
baseline estimation breslow
number of observations 4930
number of events observed 1291
partial log-likelihood -8752.37
time fit was run 2022-04-11 17:48:43 UTC
coef exp(coef) se(coef) coef lower 95% coef upper 95% exp(coef) lower 95% exp(coef) upper 95% cmp to z p -log2(p)
SeniorCitizen 0.17 1.19 0.07 0.04 0.31 1.04 1.36 0.00 2.51 0.01 6.37
TotalCharges -0.00 1.00 0.00 -0.00 -0.00 1.00 1.00 0.00 -26.70 <0.005 519.39
gender_Male -0.00 1.00 0.06 -0.11 0.11 0.89 1.11 0.00 -0.05 0.96 0.06
Partner_Yes -0.21 0.81 0.07 -0.34 -0.08 0.71 0.92 0.00 -3.16 <0.005 9.30
Dependents_Yes -0.25 0.78 0.08 -0.42 -0.09 0.66 0.91 0.00 -3.04 <0.005 8.73
MultipleLines_Yes 0.51 1.67 0.06 0.39 0.63 1.48 1.88 0.00 8.29 <0.005 53.03
OnlineSecurity_Yes -0.18 0.84 0.08 -0.33 -0.03 0.72 0.97 0.00 -2.29 0.02 5.50
OnlineBackup_Yes 0.04 1.04 0.07 -0.10 0.17 0.91 1.18 0.00 0.54 0.59 0.76
DeviceProtection_Yes 0.33 1.39 0.07 0.20 0.46 1.22 1.58 0.00 5.04 <0.005 21.03
TechSupport_Yes 0.01 1.01 0.08 -0.15 0.16 0.86 1.17 0.00 0.08 0.94 0.09
Contract_One year -1.95 0.14 0.11 -2.17 -1.73 0.11 0.18 0.00 -17.53 <0.005 226.23
Contract_Two year -4.43 0.01 0.20 -4.83 -4.02 0.01 0.02 0.00 -21.59 <0.005 340.97
PaperlessBilling_Yes 0.49 1.63 0.07 0.35 0.62 1.43 1.86 0.00 7.22 <0.005 40.83
PaymentMethod_Credit card (automatic) -0.11 0.89 0.11 -0.33 0.10 0.72 1.10 0.00 -1.05 0.29 1.76
PaymentMethod_Electronic check 0.58 1.79 0.09 0.41 0.75 1.51 2.12 0.00 6.67 <0.005 35.14
PaymentMethod_Mailed check 0.11 1.12 0.11 -0.09 0.32 0.91 1.38 0.00 1.08 0.28 1.84

Concordance 0.92
Partial AIC 17536.74
log-likelihood ratio test 3204.21 on 16 df
-log2(p) of ll-ratio test inf

The summary provide some key information:

  • p-values indicate how significant a feature is
  • exp(coef) indicate by how much the hazard will increase when the feature is increased by 1 unit
  • concordance

5 Evaluation

5.1 Plot the coefficients

fig_coef, ax_coef = plt.subplots(figsize=(12,10))
ax_coef.set_title('Survival Regression: Coefficients and Confident Intervals')
cph.plot(ax=ax_coef);
cph.plot_partial_effects_on_outcome(
    covariates='Contract_Two year', 
    values=[0, 1], 
    plot_baseline=True,
    cmap='coolwarm');
cph.plot_partial_effects_on_outcome(
    covariates='Contract_One year', 
    values=[0, 1], 
    plot_baseline=True,
    cmap='coolwarm');

5.2 Predict on the train set

surv_pred_train = cph.predict_survival_function(df_train)
surv_median_pred_train = cph.predict_median(df_train)
surv_pred_train
7923-IYJWY 3665-JATSN 3214-IYUUQ 5181-OABFK 0689-DSXGL 5630-IXDXV 0841-NULXI 4770-QAZXN 8717-VCTXJ 2480-EJWYP 0334-ZFJSR 8749-CLJXC 7576-ASEJU 6303-KFWSL 3814-MLAXC 3470-BTGQO 3566-HJGPK 1376-HHBDV 0795-XCCTE 8063-GBATB 9860-LISIZ 6469-MRVET 6618-RYATB 7107-UBYKY 5879-HMFFH 9351-LZYGF 7181-BQYBV 1579-KLYDT 6166-YIPFO 7096-UCLNH 8205-MQUGY 3923-CSIHK 3530-CRZSB 8993-IZEUX 1395-WSWXR 4274-OWWYO 9880-TDQAC 5701-YVSVF 3785-NRHYR 9565-AXSMR 5197-LQXXH 3318-NMQXL 2886-KEFUM 4776-XSKYQ 9588-YRFHY 1397-XKKWR 0679-IDSTG 0330-IVZHA 5561-NWEVX 7450-NWRTR 2664-XJZNO 3948-XHGNA 6362-QHAFM 1230-QAJDW 9804-ICWBG 4376-KFVRS 1741-WTPON 0495-ZBNGW 6171-ZTVYB 7774-OJSXI 3229-USWAR 7009-LGECI 0557-ASKVU 0311-QYWSS 9667-EQRXU 4958-GZWIY 0516-WJVXC 1563-IWQEX 9685-WKZGT 5568-DMXZS 6094-ZIVKX 7629-WFGLW 6711-FLDFB 8204-TIFGJ 2138-VFAPZ 2754-XBHTB 8350-NYMVI 8679-JOEVF 2027-CWDNU 8747-UDCOI 5802-ADBRC 5198-HQAEN 1972-XMUWV 5183-KLYEM 5821-MMEIL 6496-SLWHQ 3597-YASZG 0754-EEBDC 7398-LXGYX 6713-OKOMC 2979-SXESE 0117-LFRMW 6161-UUUTA 9944-HKVVB 7459-RRWQZ 9710-NJERN 9770-KXGQU 8082-GHXOP 6519-ZHPXP 7729-JTEEC 1410-RSCMR 0032-PGELS 4277-PVRAN 8079-XRJRS 2005-DWQZJ 6502-MJQAE 1428-GTBJJ 4373-MAVJG 2139-FQHLM 7131-ZQZNK 1852-QSWCD 7471-WNSUF 3795-CAWEX 0027-KWYKW 1218-VKFPE 6728-WYQBC 7426-GSWPO 6745-JEFZB 1047-NNCBF 3404-JNXAX 2640-PMGFL 4671-VJLCL 1450-SKCVI 4572-DVCGN 6865-JZNKO 1697-BCSHV 2612-RRIDN 1498-DQNRX 7551-DACSP 3932-CMDTD 9778-OGKQZ 4622-YNKIJ 3908-MKIMJ 9540-JYROE 8285-ABVLB 7872-BAAZR 7636-OWBPG 2521-NPUZR 6135-OZQVA 4815-GBTCD 0825-CPPQH 3352-RICWQ 9847-HNVGP 7908-QCBCA 8564-LDKFL 0813-TAXXS 1016-DJTSV 2172-EJXVF 1550-LOAHA 5153-RTHKF 9824-QCJPK 2202-CUYXZ 9767-FFLEM 8169-SAEJD 3717-OEAUQ 3336-JORSO 1302-TPUBN 1599-MMYRQ 0621-CXBKL 9853-JFZDU 9640-ZSLDC 9372-TXXPS 7024-OHCCK 2969-VAPYH 8999-XXGNS 4020-KIUDI 9074-KGVOX 2259-OUUSZ 5018-LXQQG 1568-LJSZU 8818-XYFCQ 3001-CBHLQ 7427-AUFPY 4587-VVTOX 3002-WQZWT 9896-UYMIE 4013-GUXND 0793-TWELN 5485-WUYWF 1363-TXLSL 5146-YYFRZ 0089-IIQKO 3720-DBRWL 6410-LEFEN 3649-JPUGY 1555-HAPSU 4989-LIXVT 5287-QWLKY 5150-ITWWB 9565-DJPIB 4299-OPXEJ 3806-YAZOV 6101-IMRMM 2817-LVCPP 4083-BFNYK 9820-RMCQV 5876-HZVZM 2405-LBMUW 6769-DCQLI 2233-FAGXV 2509-TFPJU 9473-CBZOP 9626-VFRGG 4312-KFRXN 0487-VVUVK 9909-IDLEK 1208-DNHLN 1074-AMIOH 5575-GNVDE 7526-IVLYU 1251-KRREG 1427-VERSM 9208-OLGAQ 5084-OOVCJ 0281-CNTZX 1231-YNDEK 4568-KNYWR 8690-UPCZI 7826-VVKWT 2612-RANWT 5811-IWXYM 8922-NPKBJ 2002-MZHWP 3541-ZNUHK 6383-ZTSIW 0840-DFEZH 6837-BJYDQ 0031-PVLZI 3389-YGYAI 9565-FLVCG 4830-FAXFM 9067-SQTNS 8095-WANWK 1080-BWSYE 0947-MUGVO 8042-RNLKO 1216-BGTSP 7765-LWVVH 5197-YPYBZ 8378-LKJAF 5650-YLIBA 1131-QQZEB 3967-KXAPS 2792-VPPET 0495-RVCBF 2186-QZEYA 7976-CICYS 3178-CIFOT 0307-BCOPK 4328-VUFWD 2311-QYMUQ 5296-BFCYD 4988-IQIGL 8328-SKJNO 8745-PVESG 0975-UYDTX 0082-OQIQY 8259-DZLIZ 2926-JEJJC 4795-UXVCJ 8577-QSOCG 7016-NVRIC 0282-NVSJS 0761-AETCS 3161-GETRM 9420-LOJKX 3812-LRZIR 1769-GRUIK 0036-IHMOT 3594-KADLU 3173-WSSUE 7850-THJMU 6969-MVBAI 5014-WUQMG 4445-ZJNMU 2725-KXXWT 8752-GHJFU 0524-IAVZO 9378-FXTIZ 9259-PACGQ 8608-OZTLB 2794-XIMMO 3165-HDOEW 2925-VDZHY 0325-XBFAC 6267-DCFFZ 6599-CEBNN 1320-REHCS 5176-OLSKT 4998-IKFSE 8468-EHYJA 4029-HPFVY 4369-NYSCF 0772-GYEQQ 2632-IVXVF 5285-MVEHD 4452-ROHMO 6906-MPARY 4160-AMJTL 3179-GBRWV 1452-UZOSF 0186-CAERR 4307-KWMXE 8957-THMOA 0670-KDOMA 3067-SVMTC 9398-MMQTO 9436-ZBZCT 7325-ENZFI 0701-RFGFI 1480-IVEVR 4385-GZQXV 9381-NDKME 2876-GZYZC 1031-IIDEO 2983-ZANRP 1379-FRVEB 0867-LDTTC 5309-TAIKL 3224-DFQNQ 4726-DLWQN 8996-ZROXE 7954-MLBUN 6522-OIQSX 3569-EDBPQ 7233-PAHHL 7639-SUPCW 9717-IOAAF 2672-OJQZP 8879-ZKJOF 5281-BUZGT 3557-HTYWR 2167-FQSTQ 4525-VZCZG 1821-BUCWY 2996-XAUVF 0214-JHPFW 6341-AEVKX 2826-DXLQO 7733-UDMTP 1530-ZTDOZ 6348-SNFUS 0263-FJTQO 3411-WLRSQ 1757-TCATG 3776-EKTKM 6260-ONULR 8015-IHCGW 5375-XLDOF 5206-XZZQI 7402-PWYJJ 9548-ZMVTX 2737-WFVYW 9194-GFVOI 3640-JQGJG 3943-KDREE 8999-BOHSE 4139-JPIAM 8540-ZQGEA 6435-VWCCY 6772-KSATR 3727-OVPRY 1200-TUZHR 8654-DHAOW 5249-QYHEX 3915-ODIYG 7234-KMNRQ 5626-MGTUK 7103-ZGVNT 8201-AAXCB 7673-LPRNY 7869-ZYDST 3143-JQEGI 8443-WVPSS 2809-ILCYT 8779-YIQQA 0930-EHUZA 6728-VOIFY 7216-KAOID 9631-XEYKE 6034-YMTOB 0301-KOBTQ 2359-KMGLI 0883-EIBTI 9531-NSBMR 6121-TNHBO 5884-GCYMI 5915-ANOEI 4195-SMMNX 9921-EZKBY 5965-GGPRW 3620-EHIMZ 6877-LGWXO 8401-EMUWF 2234-EOFPT 7602-DBTOU 0690-SRQID 3170-GWYKC 9546-KDTRB 5167-GBFRE 9530-GRMJG 9586-JGQKH 0829-XXPLX 6583-KQJLK 8372-JUXUI 0181-RITDD 2240-HSJQD 7134-HBPBS 3764-MNMOI 0827-ITJPH 5266-PFRQK 1552-CZCLL 6425-YQLLO 1849-RJYIG 4770-UEZOX 3331-HQDTW 6738-ISCBM 3834-XUIFC 1545-ACTAS 2974-GGUXS 1740-CSDJP 8853-TZDGH 2507-QZPQS 1357-BIJKI 0454-OKRCT 3340-QBBFM 1431-AIDJQ 7636-PEPNS 5649-ANRML 8022-BECSI 9135-HSWOC 2049-BAFNW 6807-SIWJI 4018-KJYUY 7544-ZVIKX 9184-GALIL 4566-NECEV 1293-HHSHJ 6719-FGEDO 1450-GALXR 5027-YOCXN 1679-JRFBR 2424-WVHPL 5913-INRQV 2342-CKIAO 3097-PYWXL 6234-PFPXL 5321-NTRKC 1455-UGQVH 7606-BPHHN 4903-UYAVB 1245-HARPS 5795-BKOYE 2267-WTPYD 4879-GZLFH 0114-IGABW 0298-XACET 3466-BYAVD 7581-EBBOU 9350-ZXYJC 6447-GORXK 3566-CAAYU 1725-MIMXW 8091-TTVAX 8975-SKGRX 4343-EJVQB 9974-JFBHQ 1775-KWJKQ 4132-KALRO 9644-KVCNC 3913-RDSJZ 2990-IAJSV 9894-QMIMJ 5170-PTRKA 8292-FRFZQ 5569-IDSEY 3594-BDSOA 4304-XUMGI 4480-QQRHC 6050-IJRHS 2740-TVLFN 7253-UVNDW 8896-RAZCR 9591-YVTEB 0866-QLSIR 8229-TNIQA 6121-VZNQB 1197-BVMVG 2800-QQUSO 4570-QHXHL 9659-ZTWSM 6582-OIVSP 8168-UQWWF 6987-XQSJT 0824-VWDPO 7017-VFULY 0334-GDDSO 6402-ZFPPI 6990-YNRIO 1150-WFARN 6198-ZFIOJ 3082-YVEKW 3494-JCHRQ 0020-INWCK 9546-CQJSU 9026-LHEVG 2495-KZNFB 3653-NCRDJ 8958-JPTRR 2207-OBZNX 2573-GYRUU 1930-QPBVZ 9978-HYCIN 0960-HUWBM 8017-LXHFA 3685-YLCMQ 6870-ECSHE 6480-YAGIY 5600-PDUJF 2239-JALAW 7291-CDTMJ 7601-DHFWZ 7694-VLBWQ 7328-ZJAJO 8393-JMVMB 9058-HRZSV 4419-UJMUS 6248-BSHKG 4612-THJBS 6698-OXETB 0853-TWRVK 3629-WEAAM 8837-VVWLQ 2717-HVIZY 5240-IJOQT 8348-JLBUG 3466-WAESX 5921-NGYRH 7780-OTDSO 4163-NCJAK 8644-XYTSV 1307-TVUFB 4884-TVUQF 9626-WEQRM 6257-DTAYD 2303-PJYHN 5093-FEGLU 1429-UYJSV 0856-NAOES 5242-UOWHD 6388-TABGU 4958-XCBDQ 4906-ZHGPK 6772-WFQRD 2692-BUCFV 7716-YTYHG 6296-DDOOR 9224-VTYID 4703-MQYKT 9492-TOKRI 4366-CTOUZ 9617-UDPEU 4817-QRJSX 9450-TRJUU 1396-QWFBJ 9330-VOFSZ 8123-QBNAZ 4054-CUMIA 6463-HHXJR 8455-HIRAQ 1678-FYZOW 5924-IFQTT 0076-LVEPS 9955-QOPOY 6954-OOYZZ 4627-MIHJH 2589-AYCRP 3951-NJCVI 7156-MXBJE 1042-HFUCW 5150-LJNSR 5324-KTGCG 3055-OYMSE 8606-OEGQZ 2388-LAESQ 3766-EJLFL 0314-TKOSI 1196-AMORA 5148-ORICT 4291-HPAXL 5650-VDUDS 1389-CXMLU 0511-JTEOY 3096-GKWEB 1704-NRWYE 5125-CNDSP 0318-ZOPWS 9798-DRYDS 1915-OAKWD 3213-VVOLG 2026-TGDHM 9025-AOMKI 6047-YHPVI 6982-UQZLY 6204-IEUXJ 7184-LRUUR 3597-MVHJT 0048-LUMLS 7294-TMAOP 4248-QPAVC 3623-FQBOX 6838-YAUVY 2478-EEWWM 3694-DELSO 9102-IAYHT 3606-TWKGI 9701-CDXHR 3096-JRDSO 2697-NQBPF 7537-RBWEA 8714-EUHJO 9391-YZEJW 1338-CECEE 6741-EGCBI 9391-LMANN 5917-HBSDW 4742-DRORA 5027-QPKTE 8617-ENBDS 4425-OWHWB 4644-PIZRT 2007-QVGAW 9766-HGEDE 7377-DMMRI 6847-KJLTS 1349-WXNGG 4359-INNWN 6821-BUXUX 3351-NGXYI 5729-KLZAR 9058-MJLZC 5399-ZIMKF 0872-JCPIB 7710-JSYOA 7479-NITWS 4302-ZYFEL 4704-ERYFC 4361-FEBGN 6563-VNPMN 0666-UXTJO 2025-JKFWI 6683-VLCTZ 4797-AXPXK 4583-PARNH 9831-BPFRI 6393-WRYZE 4872-VXRIL 3841-NFECX 8972-HJWNV 5294-DMSFH 4187-CINZD 9919-YLNNG 3472-QPRCH 9541-ZPSEA 6485-QXWWE 1977-STDKI 6096-EGVTU 7855-DIWPO 6215-NQCPY 2359-QWQUL 2937-FTHUR 7601-GNDYK 1841-YSJGV 6568-POCUI 7013-PSXHK 3694-GLTJM 4295-YURET 3442-ZHHCC 4450-YOOHP 2387-KDZQY 5673-FSSMF 9904-EHEVJ 8851-RAGOV 3668-QPYBK 6358-LYNGM 8395-ETZKQ 8327-LZKAS 8552-OBVRU 2968-SSGAA 0823-HSCDJ 7568-PODML 1088-CNNKB 3571-RFHAR 0315-LVCRK 0895-LNKRC 7762-ONLJY 4229-CZMLL 5649-VUKMC 6050-FFXES 8205-VSLRB 2351-BKRZW 4531-AUZNK 0661-WCQNQ 8562-GHPPI 9588-OZDMQ 1730-VFMWO 5366-IJEQJ 8229-BUJHX 5167-ZFFMM 3363-EWLGO 6849-WLEYG 3422-WJOYD 8838-GPHZP 3373-DIUUN 9932-WBWIK 7205-BAIAD 1729-VLAZJ 2475-MROZF 0617-AQNWT 2951-QOQTK 2466-NEJOJ 3669-OYSJI 7384-GHBPI 7693-LCKZL 7878-JGDKK 1335-NTIUC 4080-IIARD 0665-XHDJU 7426-RHZGU 2761-XECQW 6770-UAYGJ 2911-UREFD 2300-RQGOI 4806-KEXQR 9236-NDUCW 5976-JCJRH 0956-SYCWG 5766-FTRTS 9912-OMZDS 9898-KZQDZ 9773-PEQBZ 6326-MTTXK 7285-KLOTR 3703-VAVCL 0366-NQSHS 2294-DMMUS 8990-ZXLSU 3707-GNWHM 5590-BYNII 6230-BSUXY 9391-EOYLI 5895-QSXOD 8329-IBCTI 4140-MUHUG 4919-IKATY 8818-DOPVL 9084-OAYKL 9620-QJREV 7143-BQIBA 8329-GWVPJ 8272-ONJLV 3727-OWVYD 9967-ATRFS 5955-ERIHD 2432-TFSMK 1460-UZPRJ 8249-THVEC 2519-LBNQL 9298-WGMRW 8336-TAVKX 5577-OTWWW 8715-KKTFG 9681-KYGYB 1656-DRSMG 9661-MHUMO 4311-QTTAI 5996-EBTKM 5472-CVMDX 0489-WMEMG 8559-CIZFV 7852-LECYP 0708-SJDIS 0322-CHQRU 8750-QWZAJ 2073-QBVBI 0621-TSSMU 3276-HDUEG 8035-PWSEV 4631-OACRM 2946-KIQSP 7495-OOKFY 8041-TMEID 2848-YXSMW 8149-AIQCG 0447-RXSGD 1073-XXCZD 2207-QPJED 1809-DMJHQ 7997-EASSD 6925-BAYGL 5032-MIYKT 9992-UJOEL 6712-OAWRH 3014-WJKSM 1154-HYWWO 4846-WHAFZ 1307-ATKGB 3801-HMYNL 6585-WCEWR 7044-YAACC 9355-NPPFS 5840-NVDCG 2250-IVBWA 1291-CUOCY 2466-FCCPT 1833-TCXKK 3167-SNQPL 8510-BBWMU 3914-FDRHP 5977-CKHON 6733-LRIZX 1629-DQQVB 9544-PYPSJ 3853-LYGAM 6890-PFRQX 9467-ROOLM 7675-OZCZG 4250-FDVOU 2207-NHRJK 8331-ZXFOE 5171-EPLKN 7157-SMCFK 2371-JQHZZ 4226-KKDON 1877-HKBQX 1121-QSIVB 7547-EKNFS 0731-EBJQB 4310-KEDTB 2141-RRYGO 5825-XJOCM 5115-SQAAU 8433-WXGNA 5196-WPYOW 3445-HXXGF 6103-QCKFX 4999-IEZLT 1907-YLNYW 3572-UOLYZ 3911-RSNHI 3427-GGZZI 3410-YOQBQ 0945-TSONX 0274-JKUJR 5188-HGMLP 9764-REAFF 4860-YZGZM 1371-OJCEK 1591-XWLGB 4086-WITJG 6825-UYPFK 3703-KBKZP 3221-CJMSG 4007-NHVHI 2482-CZGBB 1751-NCDLI 5208-HFSBT 7998-WNZEM 3966-HRMZA 3079-BCHLN 2200-DSAAL 3413-DHLPB 3629-ZNKXA 1135-LMECX 0030-FNXPP 1045-LTCYT 6963-EZQEE 3354-OADJP 1075-BGWOH 7353-YOWFP 8642-GVWRF 5740-YHGTW 8295-KMENE 5067-WJEUN 7394-LWLYN 2265-CYWIV 7693-QPEFS 5844-QVTAT 5522-JBWMO 2782-JEEBU 0277-ORXQS 9581-GVBXT 6771-XWBDM 3671-SHRSP 5458-CQJTA 6855-VLGOS 0699-NDKJM 8582-KRHPJ 8104-OSKWT 7663-CUXZB 0168-XZKBB 7625-XCQRH 8861-HGGKB 5797-APWZC 6284-KMNUF 2790-XUYMV 1170-SASML 0225-ZORZP 7789-CRUVC 6762-QVYJO 1389-WNUIB 8857-CUPFQ 0221-WMXNQ 3223-WZWJM 3800-LYTRK 0487-RPVUM 5955-EPOAZ 1768-HNVGJ 5915-DGNVC 3498-LZGQZ 6017-PPLPX 7008-LZVOZ 7665-TOALD 5804-JMYIO 1518-VOWAV 3768-NLUBH 6360-SVNWV 7245-JMTTQ 7010-ZMVBF 1355-KUSBG 8263-OKETD 2344-JMOGN 5016-IBERQ 9761-XUJWD 6365-HITVU 8944-AILEF 6994-KERXL 0164-XAIRP 9830-ECLEN 2592-SEIFQ 4713-LZDRV 1345-ZUKID 9371-BITHB 7110-BDTWG 1325-USMEC 5406-KGRMX 5981-ZVXOT 1626-ERCMM 6331-LWDTQ 7952-OBOYL 1069-QJOEE 3583-KRKMD 6827-IEAUQ 7216-EWTRS 6569-KTMDU 1000-AJSLD 8132-YPVBX 9931-KGHOA 8414-OOEEL 5049-GLYVG 1097-FSPVW 9958-MEKUC 4114-QMKVN 5669-SRAIP 3588-WSTTJ 2511-ALLCS 0810-DHDBD 4737-HOBAX 6729-GDNGC 3780-DDGSE 2995-YWTCD 0318-QUUOB 8190-ZTQFB 1373-ORVIZ 5519-YLDGW 9499-XPZXM 2878-RMWXY 2272-UOINI 0871-OPBXW 2731-GJRDG 4680-KUTAJ 9530-EHPOH 5712-AHQNN 0902-RFHOF 2452-KDRRH 4011-ARPHK 8245-UMPYT 7235-NXZCP 4992-LTJNE 9711-FJTBX 1205-WNWPJ 5233-AOZUF 5478-JJVZK 5088-QZLRL 3748-FVMZZ 0927-LCSMG 9405-GPBBG 3996-ZNWYK 2324-EFHVG 2560-QTSBS 0048-PIHNL 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6.973648e-01 8.751719e-01 0.982474 0.999904 0.999988 0.999986 8.903288e-01 8.701161e-01 8.287699e-01 9.257126e-01 0.999997 9.415405e-01 7.092120e-01 0.998385 0.999995 9.120398e-01 7.822853e-01 0.999904 0.999976 0.986594 0.999795 8.169698e-01 0.999916 0.994009 8.513757e-01 0.999693 8.892970e-01 0.994632 0.996100 0.999977 0.999474 8.675722e-01 6.572809e-01 0.996052 0.976094 8.737131e-01 9.717643e-01 0.999238 8.611043e-01 0.987376 0.994588 0.990303 8.595049e-01 0.994239 8.795518e-01 0.999993 0.988499 0.999720 5.910840e-01 7.769099e-01 9.595862e-01 6.263980e-01 8.496276e-01 0.999975 7.845050e-01 8.288980e-01 0.985430 0.987274 7.890650e-01 8.865554e-01 0.999898 0.977536 6.968877e-01 8.353305e-01 0.999515 0.999890 7.218900e-01 0.983194 0.992442 0.999993 0.994554 0.999988 7.880924e-01 9.620639e-01 0.999446 0.999559 9.633682e-01 0.999976 8.708929e-01 0.999559 7.900602e-01 7.952412e-01 0.999993 0.989313 9.244398e-01 0.999990 0.999991 9.264416e-01 0.999710 8.460849e-01 5.072054e-01 8.468754e-01 7.245906e-01 8.976324e-01 7.669185e-01 0.999977 9.005989e-01 7.679153e-01 8.262018e-01 9.588323e-01 9.496059e-01 7.950006e-01 0.999839 0.987493 0.999954 0.999969 9.141844e-01 0.999998 0.999720 0.998533 9.341013e-01 0.998397 0.999991 0.999930 0.999996 0.992461 7.228456e-01 8.556862e-01 0.994210 0.985066 5.765493e-01 8.869857e-01 7.878737e-01 7.214886e-01 7.753054e-01 9.641658e-01 0.999873 9.464513e-01 9.311735e-01 0.981363 8.734647e-01 9.729431e-01 7.684034e-01 9.077133e-01 0.999972 7.759382e-01 0.999371 9.479830e-01 7.661353e-01 0.999136 9.009145e-01 0.997966 8.202210e-01 8.864584e-01 0.990616 0.998133 8.489433e-01 0.993784 0.998980 8.459607e-01 0.999771 5.687016e-01 0.999955 0.998933 0.993516 6.643001e-01 0.999617 8.641231e-01 0.978510 0.999012 8.459288e-01 0.999991 0.982263 6.132982e-01 9.170459e-01 7.756248e-01 9.044640e-01 0.991478 0.975693 9.247542e-01 0.999978 0.999963 0.997381 0.999998 9.191596e-01 7.391866e-01 9.262491e-01 7.465988e-01 9.561465e-01 5.193570e-01 8.539328e-01 8.460307e-01 0.999986 9.111561e-01 0.992706 9.117329e-01 0.999532 5.593672e-01 0.993108 0.999050 0.998347 8.467111e-01 9.669887e-01 7.717222e-01 8.622769e-01 0.999987 8.829662e-01 0.999150 0.998960 0.998252 0.981616 0.999996 0.995266 0.999542 0.992027 8.754439e-01 9.484708e-01 0.999994 3.940403e-01 0.997761 7.601531e-01 0.999514 0.999984 0.997914 0.999901 0.998908 0.997930 0.999353 0.976865 7.702963e-01 0.996146 0.999544 0.999977 0.999176 0.974237 0.998227 6.209557e-01 8.376882e-01 9.707873e-01 0.983290 0.998053 8.335520e-01 7.592157e-01 0.994745 6.106390e-01 8.288040e-01 8.386386e-01 0.990974 9.123356e-01 0.990908 0.999901 8.907052e-01 0.999979 6.210199e-01 6.455061e-01 9.708110e-01 6.593960e-01 0.999989 0.999020 0.996917 0.997045 7.127282e-01 0.999753 0.998993 0.999604 0.997037 8.305238e-01 0.999972 0.999066 0.984491 0.999967 0.999983 0.999540 0.986717 7.658148e-01 0.999867 0.985569 0.999951 9.688260e-01 7.537750e-01 0.999961 8.871838e-01 0.999913 8.469957e-01 8.044690e-01 0.999979 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8.149681e-01 0.999983 8.923858e-01 0.991082 8.930772e-01 0.999427 4.911911e-01 0.991572 0.998838 0.997977 8.157702e-01 9.597538e-01 7.282546e-01 8.341599e-01 0.999984 8.587177e-01 0.998959 0.998728 0.997862 0.977550 0.999995 0.994210 0.999439 0.990252 8.497739e-01 9.373109e-01 0.999993 3.199285e-01 0.997261 7.149171e-01 0.999405 0.999981 0.997447 0.999879 0.998664 0.997468 0.999208 0.971763 7.266082e-01 0.995286 0.999441 0.999972 0.998991 0.968565 0.997830 5.581657e-01 8.051449e-01 9.643694e-01 0.979590 0.997618 8.002826e-01 7.138384e-01 0.993573 5.468386e-01 7.947078e-01 8.062628e-01 0.988966 8.937997e-01 0.988885 0.999879 8.679370e-01 0.999975 5.582363e-01 5.852889e-01 9.643982e-01 6.007376e-01 0.999987 0.998801 0.996228 0.996385 6.607229e-01 0.999698 0.998768 0.999515 0.996375 7.967263e-01 0.999965 0.998857 0.981054 0.999959 0.999979 0.999437 0.983769 7.214385e-01 0.999837 0.982369 0.999940 9.619857e-01 7.075835e-01 0.999952 8.637398e-01 0.999894 8.161058e-01 7.662480e-01 0.999974 0.999468 7.433942e-01 8.385650e-01 9.502975e-01 8.669766e-01 0.997911 0.991782 0.989834 0.989134 8.105820e-01 6.732902e-01 6.033419e-01 8.815839e-01 8.556646e-01 9.165868e-01 0.998971 0.999900 0.999430 0.999996 6.023753e-01 6.455794e-01 6.694948e-01 0.999982 0.982095 0.998961 8.026421e-01 7.849631e-01 0.998682 0.999962 9.508920e-01 9.145399e-01 0.999227 0.999668 7.315686e-01 8.767048e-01 0.999599 0.998874 7.622296e-01 7.714849e-01 0.999070 0.999945 7.218306e-01 6.079211e-01 0.986215 9.420793e-01 7.308659e-01 0.999314 0.996040 7.041678e-01 0.999379 0.973328 0.999954 9.109851e-01 9.634205e-01 0.997150 0.999897 8.584443e-01
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68.0 0.998804 1.013692e-23 0.983017 0.978646 0.570246 0.051984 2.572138e-41 3.279588e-11 0.017491 0.910024 0.698218 1.292678e-16 0.813522 0.003969 8.979407e-05 6.793993e-06 5.555331e-26 2.934170e-06 0.856637 2.326184e-11 1.410088e-09 2.098994e-04 1.121098e-57 1.274214e-20 0.998156 0.007495 1.963666e-47 1.124427e-57 0.984273 1.292678e-16 6.705692e-14 0.014983 7.941297e-15 5.085362e-10 0.749987 2.319026e-41 0.977507 3.371988e-19 1.275758e-16 0.897194 0.994382 1.092685e-33 0.946413 0.146706 0.794460 0.007798 2.916578e-30 2.729592e-16 3.443704e-50 5.704085e-40 0.999529 2.509799e-07 0.658419 2.973508e-37 4.428064e-40 0.999641 0.998467 3.118968e-39 0.057355 0.124168 0.461095 1.206220e-16 0.077529 3.326684e-27 6.312013e-25 2.589171e-17 3.357838e-17 4.950875e-14 2.078632e-33 2.429585e-27 0.746226 0.887134 9.182282e-44 3.912372e-16 0.697194 4.224993e-13 1.032489e-05 4.925239e-13 5.376332e-07 0.906134 0.884995 0.594218 0.984784 0.999266 0.888274 1.144269e-92 0.998544 0.769764 0.055133 4.586689e-11 3.885631e-08 1.076089e-03 3.742426e-40 9.127762e-34 3.996485e-14 0.808436 0.914420 1.178635e-07 0.999508 5.095564e-25 2.041960e-10 3.979872e-08 1.703981e-13 0.926548 0.790977 4.321001e-40 2.095692e-21 5.897109e-09 1.048330e-05 0.989707 0.823377 0.192961 0.999523 4.701874e-09 5.434706e-09 6.931258e-15 0.774375 0.028408 4.382973e-04 0.081524 2.223697e-08 0.985380 0.080506 1.025035e-11 7.181214e-06 0.192009 1.459146e-31 1.404836e-08 1.535721e-18 1.533224e-06 0.997889 0.938299 0.911109 9.595734e-17 0.916359 0.300644 4.588782e-19 0.999407 1.975595e-41 7.200398e-32 0.919480 0.792467 0.018236 7.986406e-21 0.706123 0.969855 1.683807e-30 0.955904 0.065197 6.685709e-04 0.094808 9.745599e-55 0.010666 0.991574 3.966606e-15 3.146401e-05 0.877933 0.958756 0.834526 8.097000e-05 0.826855 0.987814 3.429062e-72 0.892949 0.983549 7.132417e-08 0.016399 4.669865e-04 0.044193 0.997677 0.094311 2.424893e-05 0.866536 7.373672e-51 2.265493e-12 0.980036 1.324995e-16 1.198387e-44 4.570954e-19 1.130381e-38 0.885033 0.179641 0.988302 1.726457e-18 0.994133 0.919913 0.453230 0.943306 3.724904e-62 2.294860e-16 0.782468 3.741838e-16 4.034802e-11 0.867877 0.792541 1.311899e-07 8.565172e-06 0.769562 0.919214 2.867055e-25 0.914290 0.098061 0.065339 0.878821 1.738200e-11 3.852819e-04 0.063939 0.789488 0.324861 1.810454e-25 9.445487e-42 0.513246 1.080948e-19 1.403528e-05 0.841484 0.775133 0.975591 0.489604 0.854588 7.414894e-20 0.943253 0.879480 1.180637e-16 0.247348 0.873423 7.308636e-43 0.006660 3.011558e-10 1.099437e-57 0.988174 5.823514e-19 0.275253 9.335928e-12 0.887681 5.971854e-12 0.181706 0.340238 0.997780 0.747199 0.135895 1.898893e-07 0.876693 0.998627 7.855642e-40 7.574551e-40 2.458476e-09 3.697034e-20 0.011408 0.019096 0.311901 5.537985e-06 0.821850 2.205379e-78 0.256280 0.348406 0.245006 9.289217e-05 0.675619 1.128788e-03 0.061155 0.791650 0.615785 3.854496e-15 4.009330e-26 0.013200 9.026143e-06 0.686665 2.141091e-14 0.957743 8.413596e-11 0.939324 0.866395 2.648167e-56 0.819058 1.468653e-16 4.882876e-32 0.931198 4.472925e-06 0.720510 1.177614e-04 0.596739 0.399204 1.345887e-16 0.998704 1.151985e-30 2.625213e-07 0.005689 0.232941 0.987365 2.052528e-04 0.460381 2.647767e-22 0.844238 8.157226e-30 0.999495 0.971883 0.717583 0.733686 3.021128e-27 1.334979e-06 0.999179 0.991275 1.192856e-19 0.891141 1.225712e-26 0.903863 1.133783e-11 4.689105e-25 0.212518 0.409899 0.255972 1.232688e-13 6.747723e-13 3.548594e-24 3.685044e-40 3.505346e-14 3.190763e-16 6.580225e-15 1.264681e-14 0.951921 0.015867 0.226310 0.966696 0.982558 5.050208e-11 0.994824 1.583939e-07 0.692759 2.606494e-16 0.635244 1.134042e-04 0.972450 0.995786 0.952254 0.826629 0.942984 0.987710 0.966672 2.861708e-10 3.545242e-05 0.030141 0.989816 0.978174 1.051480e-19 0.219595 4.034661e-49 1.361011e-25 0.999252 0.999172 0.283322 0.790619 0.998362 1.210828e-39 0.977639 3.292694e-06 3.025109e-08 1.975694e-07 0.003359 0.898482 0.996271 5.413741e-55 0.215386 4.045379e-28 2.621249e-26 0.738599 0.631537 7.197889e-16 9.081178e-07 0.953578 2.381888e-08 0.006243 6.331203e-06 3.953879e-06 2.989093e-15 2.403577e-03 0.088111 0.366647 0.953842 0.235191 5.588787e-22 2.940633e-19 0.689276 2.797775e-14 1.697349e-16 0.015327 0.903366 6.337178e-16 0.995006 1.988230e-16 0.014145 0.999487 0.894870 1.241715e-07 0.043828 0.997426 0.903647 1.737650e-25 0.793594 1.392876e-16 3.460655e-26 0.991964 0.997985 8.323946e-06 6.055591e-14 2.230525e-51 0.999271 0.008242 0.998296 0.854856 0.732942 0.193337 8.872733e-04 0.971395 2.206016e-23 7.283476e-39 0.007954 0.169636 3.208474e-26 0.938180 0.016075 1.842701e-23 0.998845 7.384824e-13 0.784901 0.962567 0.387944 0.969377 0.538690 1.753100e-31 0.026766 0.878822 1.032558e-22 0.549370 1.707425e-07 0.950665 8.956773e-06 3.299572e-40 0.065486 0.998226 6.514685e-28 0.977208 0.993128 1.440375e-29 0.773309 2.323671e-16 3.018602e-25 0.011000 0.994617 1.123856e-31 0.857285 1.955765e-12 0.756758 0.900787 0.597169 0.075434 0.988828 0.960001 7.114729e-10 0.800430 0.419330 1.233396e-05 0.964530 8.965171e-21 0.953469 0.999402 4.285439e-09 0.859953 0.998774 1.284882e-31 0.926305 0.878908 0.997949 0.126541 0.481181 0.409154 0.982216 0.108568 0.969328 2.024307e-23 0.973690 1.716199e-67 0.947017 0.853208 2.288767e-05 0.147246 7.924689e-23 1.152258e-26 1.149174e-64 0.026930 6.742573e-30 0.969129 2.653204e-16 6.736239e-21 0.020405 0.882117 5.195844e-12 1.547430e-70 7.296424e-05 0.999055 0.998869 1.050718e-30 0.849383 4.482575e-40 0.998164 1.386389e-82 4.120529e-20 1.068647e-04 0.067740 0.025446 1.005365e-16 0.121267 0.999468 0.852144 0.997724 2.651603e-07 0.888278 0.005813 0.946312 1.229287e-16 0.921292 8.788893e-24 0.814926 0.971368 0.928221 1.267379e-16 0.756770 0.997462 0.331650 0.856930 0.696190 0.999292 3.756810e-15 1.447802e-12 1.389564e-11 1.986233e-78 0.949628 2.000471e-18 0.569108 1.377248e-10 0.018555 2.085840e-04 0.004197 0.995463 1.035452e-20 0.998926 6.385574e-12 0.868412 0.015001 3.152730e-16 0.885623 0.871591 1.079501e-05 0.604427 0.063343 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7.019043e-19 0.996535 7.907652e-11 0.160399 9.263158e-11 0.889474 8.381861e-64 0.177452 0.788503 0.661220 8.581422e-19 2.266227e-04 7.325748e-29 8.155630e-17 0.996699 3.060756e-14 0.808390 0.770971 0.645755 0.009670 0.998989 0.305331 0.891699 0.135158 3.604782e-15 1.802659e-06 0.998580 7.650770e-102 0.570997 1.678216e-30 0.885561 0.996070 0.593236 0.975521 0.761018 0.595715 0.850536 0.002875 4.613653e-29 0.380867 0.892139 0.994373 0.813716 0.001466 0.641639 1.838316e-52 5.892933e-20 6.039181e-04 0.014803 0.614278 1.709611e-20 1.232731e-30 0.267877 2.788505e-54 4.098941e-21 7.824233e-20 0.103655 1.092725e-10 0.101922 0.975562 2.712153e-13 0.994842 1.886435e-52 2.981343e-48 6.076084e-04 6.108364e-46 0.997341 0.782579 0.462084 0.477156 1.700051e-37 0.940193 0.777434 0.905647 0.476225 6.882374e-21 0.992943 0.791670 0.020087 0.991722 0.995671 0.891388 0.035327 1.072809e-29 0.967334 0.026406 0.987718 3.642455e-04 2.041739e-31 0.990195 1.007420e-13 0.978558 9.333526e-19 2.382237e-24 0.994677 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7.731606e-74 0.021786 2.289723e-10 8.800977e-28 5.247550e-18 0.975942 0.964230 3.782268e-07 4.012925e-59 2.501402e-22 0.001366 0.964855 0.995655 0.994615 1.513570e-19 2.881034e-23 3.733821e-31 3.119463e-13 0.998849 1.740867e-10 2.151185e-56 0.547277 0.998127 1.211478e-15 1.660919e-40 0.964650 0.991216 0.006508 0.926352 1.773569e-33 0.969069 0.106274 8.551537e-27 0.891705 9.821011e-20 0.134292 0.232726 0.991598 0.821754 9.664764e-24 1.029973e-68 0.228654 0.000120 1.342520e-22 2.289090e-05 0.752515 5.926997e-25 0.008743 0.132065 0.026382 2.962368e-25 0.115857 1.610550e-21 0.997503 0.013362 0.900780 6.527794e-86 1.268402e-41 2.072630e-07 1.644147e-76 3.972296e-27 0.990872 4.779643e-40 3.955404e-31 0.004190 0.008414 4.153258e-39 3.103967e-20 0.962636 0.000208 3.108678e-59 7.072958e-30 0.834355 0.959869 1.596842e-53 0.001795 0.059006 0.997450 0.130398 0.995644 2.621589e-39 5.423784e-07 0.813143 0.848425 8.990710e-07 0.991064 4.019018e-23 0.848182 6.646595e-39 7.612794e-38 0.997433 0.018169 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2.203712e-14 1.094727e-49 3.872126e-13 4.526740e-48 5.429223e-08 7.175550e-107 2.617659e-26 8.160708e-28 0.994834 8.438451e-16 0.065168 1.068541e-15 0.839651 7.576739e-95 0.075772 0.701478 0.539420 1.101452e-27 3.643566e-06 1.041923e-42 9.847089e-25 0.995078 6.834135e-21 0.728040 0.678332 0.520703 0.000986 0.998492 0.170293 0.842787 0.050475 2.808932e-22 2.684722e-09 0.997882 1.312896e-151 0.433370 3.721224e-45 0.834144 0.994142 0.458796 0.963695 0.665307 0.461659 0.785399 0.000161 5.226358e-43 0.236835 0.843408 0.991616 0.735209 0.000059 0.515759 6.341567e-78 2.024151e-29 1.572902e-05 0.001862 0.483289 3.193776e-30 2.348370e-45 0.140084 1.224288e-80 3.791649e-31 3.089891e-29 0.033971 1.367277e-15 0.033127 0.963754 1.772084e-19 0.992313 6.590848e-78 1.213896e-71 1.587265e-05 3.414525e-68 0.996035 0.693628 0.316017 0.331520 1.361248e-55 0.912085 0.686835 0.862533 0.330555 8.216063e-31 0.989489 0.705686 0.002935 0.987673 0.993547 0.842348 0.006816 5.927637e-44 0.951651 0.004415 0.981729 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0.634332 3.070828e-35 1.014642e-93 0.000007 0.138233 0.661532 0.716304 7.272050e-42 9.554064e-100 0.005614 8.783104e-14 2.275504e-37 3.949581e-24 0.967556 0.951863 2.004224e-09 8.126372e-80 5.537352e-30 0.000132 0.952700 0.994119 0.992714 3.242608e-26 2.965170e-31 6.161052e-42 1.154639e-17 0.998442 6.059762e-14 4.038744e-76 0.442016 0.997464 6.271862e-21 1.330202e-54 0.952425 0.988122 0.001093 0.901577 4.395987e-45 0.958340 0.048020 4.949130e-36 0.856213 1.805022e-26 0.065927 0.138829 0.988638 0.766530 6.754573e-32 8.330993e-93 0.135549 0.000005 2.383791e-30 5.191583e-07 0.680388 1.540378e-33 0.001630 0.064451 0.007276 6.021421e-34 0.053977 6.897606e-29 0.996619 0.002896 0.868036 4.248972e-116 4.082914e-56 8.874529e-10 2.295540e-103 1.751954e-36 0.987658 5.567161e-54 6.661385e-42 0.000602 0.001548 1.040810e-52 3.792950e-27 0.949734 0.000010 5.750660e-80 3.309653e-40 0.782493 0.946038 3.119315e-72 0.000191 0.021644 0.996548 0.063351 0.994105 5.581321e-53 3.265667e-09 0.755672 0.800418 6.475050e-09 0.987916 4.654274e-31 0.800108 1.967644e-52 5.347392e-51 0.996525 0.004390 5.761926e-18 0.994869 0.995563 1.718653e-17 0.863828 2.121742e-37 1.115346e-149 3.400960e-37 2.057723e-71 2.011644e-24 5.899554e-59 0.988415 1.065382e-23 1.137182e-58 1.284314e-42 5.965927e-10 4.508654e-12 4.589287e-51 0.921989 0.001732 0.976832 0.984248 2.054707e-20 0.999146 0.868015 0.476195 1.101218e-15 0.444543 0.995327 0.965074 0.997806 0.021858 6.085875e-72 6.348532e-35 0.053182 0.000499 1.463056e-121 4.846752e-27 4.851220e-53 2.355018e-72 1.436628e-56 9.836535e-09 0.937940 8.393050e-13 2.254560e-16 0.000074 2.064702e-30 9.578809e-07 1.567584e-58 5.676123e-22 0.986061 2.169495e-56 0.727675 1.899959e-12 3.520543e-59 0.646214 1.271644e-23 0.357500 3.269778e-44 3.588862e-27 0.008533 0.388963 1.166133e-36 0.042829 0.597068 1.970068e-37 0.890809 1.439050e-124 0.977613 0.583156 0.037370 1.784539e-90 0.823855 9.026587e-33 0.000017 0.606875 1.932946e-37 0.995476 0.000118 5.289869e-108 9.965886e-20 1.769007e-56 9.272172e-23 0.013242 0.000004 6.841820e-18 0.988931 0.981397 0.265799 0.999136 3.189160e-19 4.891637e-67 1.547319e-17 7.564053e-65 1.446101e-10 1.747081e-144 2.252017e-35 2.054236e-37 0.993009 3.843175e-21 0.024761 5.291163e-21 0.789228 3.362535e-128 0.030370 0.618653 0.433443 3.083469e-37 4.308575e-08 1.383298e-57 3.063570e-33 0.993339 4.884774e-28 0.650592 0.591169 0.413200 0.000085 0.997957 0.090941 0.793223 0.017518 6.478952e-30 2.463881e-12 0.997133 4.480593e-205 0.322236 6.707908e-61 0.782226 0.992074 0.348105 0.951148 0.575848 0.351050 0.720965 0.000007 5.433747e-58 0.142159 0.794014 0.988662 0.659283 0.000002 0.407896 2.793589e-105 1.374805e-39 3.123138e-07 0.000201 0.373509 1.127523e-40 3.596035e-61 0.069807 5.888406e-109 6.290640e-42 2.438024e-39 0.010247 7.388500e-21 0.009903 0.951228 4.014612e-26 0.989604 2.943344e-105 9.000143e-97 3.161826e-07 4.223266e-92 0.994634 0.609296 0.210100 0.224179 4.914084e-75 0.882822 0.601228 0.818496 0.223296 1.792826e-41 0.985791 0.623684 0.000372 0.983341 0.991270 0.792664 0.001164 2.849683e-59 0.935085 0.000646 0.975335 1.124160e-07 9.500506e-63 0.980284 5.425446e-27 0.957142 3.654063e-37 1.818059e-48 0.989271 0.802639 6.765562e-54 2.695495e-32 7.226065e-10 2.541642e-26 0.421769 0.033139 0.014716 0.010991 2.213238e-38 1.174703e-71 2.519446e-91 2.517744e-23 1.122518e-28 2.413362e-16 0.653761 0.959369 0.790252 0.998229 1.299603e-91 3.419521e-79 1.138427e-72 0.992592 0.000576 0.651066 3.801880e-40 3.858562e-44 0.580213 0.984303 9.354549e-10 9.588656e-17 0.726821 0.871751 9.016146e-57 2.546208e-24 0.847553 0.628071 2.073989e-49 3.026186e-47 0.681042 0.977577 3.566286e-59 5.715706e-90 0.003244 2.002151e-11 6.063390e-57 0.753276 0.194299 1.288556e-63 0.773883 0.000014 0.981121 1.920563e-17 2.080010e-07 0.307784 0.958194 4.282976e-28

73 rows × 4930 columns

kmf.survival_function_
Estimate for Average Customer
timeline
0.0 1.000000
1.0 0.945961
2.0 0.927835
3.0 0.913725
4.0 0.901045
... ...
68.0 0.626858
69.0 0.620135
70.0 0.609657
71.0 0.602781
72.0 0.592790

73 rows × 1 columns

import plotly.graph_objs as go
fig = go.Figure()
buttons = []
fig.add_trace(
    go.Scatter(
        x=kmf.survival_function_.index, 
        #.y=kmf.survival_function_['KM_estimate'],
        y=kmf.survival_function_,
        marker_color="black",
        #.name='KM estimate'
    )
)
buttons.append(dict(method='restyle', 
                    #.label='KM Estimate', 
                    #.args=[{'y': [kmf.survival_function_['KM_estimate'].values]}]))
                    args=[{'y': [kmf.survival_function_.values]}]))
for customerID in surv_pred_train.columns[0:50]:
    #.color = 'red' if df_train_clean.loc[customerID, 'status'] else 'blue'
    color = 'red' if df_train.loc[customerID, 'Churn'] else 'blue'
    customer_probabilities = surv_pred_train[customerID]
    fig.add_trace(
        go.Scatter(
            x=surv_pred_train.index, 
            y=customer_probabilities,
            marker_color=color,
            name=customerID
        )
    )
    buttons.append(dict(method='restyle',
                        label=customerID,
                        args=[{'y': [customer_probabilities.values]}])
                  )
updatemenu = [{}]
updatemenu[0]['buttons'] = buttons
updatemenu[0]['direction'] = 'down'
updatemenu[0]['showactive'] = True

# update layout and show figure
fig.update_layout(updatemenus=updatemenu)
surv_pred_train.columns[0:5]
Index(['7923-IYJWY', '3665-JATSN', '3214-IYUUQ', '5181-OABFK', '0689-DSXGL'], dtype='object')
surv_median_pred_train
7923-IYJWY     inf
3665-JATSN    13.0
3214-IYUUQ     inf
5181-OABFK     inf
0689-DSXGL    70.0
              ... 
7503-QQRVF    31.0
9971-ZWPBF    46.0
8612-GXIDD    70.0
8065-QBYTO     inf
7764-BDPEE    23.0
Name: 0.5, Length: 4930, dtype: float64
patient_name = surv_pred_train.columns[19]
pred_median = surv_median_pred_train
ax = surv_pred_train[patient_name].plot()
kmf.plot(ax=ax, label='KM Estimator')
plt.vlines(df_train.loc[patient_name, 'tenure'], ymin=0, ymax=1)
plt.vlines(surv_median_pred_train[patient_name], ymin=0, ymax=1, color='red')
<matplotlib.collections.LineCollection at 0x7fa310cd6ed0>

5.3 Predict on the validation set

surv_pred_valid = cph.predict_survival_function(df_valid)
surv_median_pred_valid = cph.predict_median(df_valid)
patient_name = surv_pred_valid.columns[100]
pred_median = surv_median_pred_valid
ax = surv_pred_valid[patient_name].plot()
kmf.plot(ax=ax, label='KM Estimator')
plt.vlines(df_valid.loc[patient_name, 'tenure'], ymin=0, ymax=1)
plt.vlines(surv_median_pred_valid[patient_name], ymin=0, ymax=1, color='red');
ax = surv_pred_valid.iloc[:, 0:6].plot();
kmf.plot(ax=ax, label='KM Estimator');
from lifelines.utils import concordance_index
concordance_index(df_valid['tenure'], -cph.predict_partial_hazard(df_valid), df_valid['Churn'])
0.9264000512153804

5.5 Check assumptions

%%time
cph.check_assumptions(df_train.loc[:, features_keep], p_value_threshold=0.05, show_plots=True)
The ``p_value_threshold`` is set at 0.05. Even under the null hypothesis of no violations, some
covariates will be below the threshold by chance. This is compounded when there are many covariates.
Similarly, when there are lots of observations, even minor deviances from the proportional hazard
assumption will be flagged.

With that in mind, it's best to use a combination of statistical tests and visual tests to determine
the most serious violations. Produce visual plots using ``check_assumptions(..., show_plots=True)``
and looking for non-constant lines. See link [A] below for a full example.

null_distribution chi squared
degrees_of_freedom 1
model <lifelines.CoxPHFitter: fitted with 4930 total...
test_name proportional_hazard_test
test_statistic p -log2(p)
Contract_One year km 59.60 <0.005 46.29
rank 44.75 <0.005 35.38
Contract_Two year km 53.45 <0.005 41.78
rank 36.86 <0.005 29.55
Dependents_Yes km 0.36 0.55 0.87
rank 0.02 0.89 0.16
DeviceProtection_Yes km 0.42 0.51 0.96
rank 1.64 0.20 2.32
MultipleLines_Yes km 4.20 0.04 4.63
rank 14.12 <0.005 12.51
OnlineBackup_Yes km 0.14 0.71 0.50
rank 0.02 0.88 0.19
OnlineSecurity_Yes km 3.64 0.06 4.15
rank 4.69 0.03 5.04
PaperlessBilling_Yes km 0.08 0.78 0.36
rank 0.48 0.49 1.04
Partner_Yes km 0.04 0.85 0.23
rank 0.46 0.50 1.00
PaymentMethod_Credit card (automatic) km 6.36 0.01 6.42
rank 4.97 0.03 5.27
PaymentMethod_Electronic check km 4.88 0.03 5.20
rank 2.27 0.13 2.92
PaymentMethod_Mailed check km 3.41 0.06 3.95
rank 0.52 0.47 1.09
SeniorCitizen km 8.42 <0.005 8.07
rank 4.85 0.03 5.18
TechSupport_Yes km 4.43 0.04 4.82
rank 7.02 0.01 6.95
TotalCharges km 623.93 <0.005 455.04
rank 385.43 <0.005 282.65
gender_Male km 1.52 0.22 2.20
rank 0.47 0.49 1.02

1. Variable 'SeniorCitizen' failed the non-proportional test: p-value is 0.0037.

   Advice: with so few unique values (only 2), you can include `strata=['SeniorCitizen', ...]` in
the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


2. Variable 'TotalCharges' failed the non-proportional test: p-value is <5e-05.

   Advice 1: the functional form of the variable 'TotalCharges' might be incorrect. That is, there
may be non-linear terms missing. The proportional hazard test used is very sensitive to incorrect
functional forms. See documentation in link [D] below on how to specify a functional form.

   Advice 2: try binning the variable 'TotalCharges' using pd.cut, and then specify it in
`strata=['TotalCharges', ...]` in the call in `.fit`. See documentation in link [B] below.

   Advice 3: try adding an interaction term with your time variable. See documentation in link [C]
below.


   Bootstrapping lowess lines. May take a moment...


3. Variable 'MultipleLines_Yes' failed the non-proportional test: p-value is 0.0002.

   Advice: with so few unique values (only 2), you can include `strata=['MultipleLines_Yes', ...]`
in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


4. Variable 'OnlineSecurity_Yes' failed the non-proportional test: p-value is 0.0303.

   Advice: with so few unique values (only 2), you can include `strata=['OnlineSecurity_Yes', ...]`
in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


5. Variable 'TechSupport_Yes' failed the non-proportional test: p-value is 0.0081.

   Advice: with so few unique values (only 2), you can include `strata=['TechSupport_Yes', ...]` in
the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


6. Variable 'Contract_One year' failed the non-proportional test: p-value is <5e-05.

   Advice: with so few unique values (only 2), you can include `strata=['Contract_One year', ...]`
in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


7. Variable 'Contract_Two year' failed the non-proportional test: p-value is <5e-05.

   Advice: with so few unique values (only 2), you can include `strata=['Contract_Two year', ...]`
in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


8. Variable 'PaymentMethod_Credit card (automatic)' failed the non-proportional test: p-value is 0.0117.

   Advice: with so few unique values (only 2), you can include `strata=['PaymentMethod_Credit card
(automatic)', ...]` in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


9. Variable 'PaymentMethod_Electronic check' failed the non-proportional test: p-value is 0.0271.

   Advice: with so few unique values (only 2), you can include `strata=['PaymentMethod_Electronic
check', ...]` in the call in `.fit`. See documentation in link [E] below.

   Bootstrapping lowess lines. May take a moment...


---
[A]  https://lifelines.readthedocs.io/en/latest/jupyter_notebooks/Proportional%20hazard%20assumption.html
[B]  https://lifelines.readthedocs.io/en/latest/jupyter_notebooks/Proportional%20hazard%20assumption.html#Bin-variable-and-stratify-on-it
[C]  https://lifelines.readthedocs.io/en/latest/jupyter_notebooks/Proportional%20hazard%20assumption.html#Introduce-time-varying-covariates
[D]  https://lifelines.readthedocs.io/en/latest/jupyter_notebooks/Proportional%20hazard%20assumption.html#Modify-the-functional-form
[E]  https://lifelines.readthedocs.io/en/latest/jupyter_notebooks/Proportional%20hazard%20assumption.html#Stratification

[[<matplotlib.axes._subplots.AxesSubplot at 0x7fa30f24de90>,
  <matplotlib.axes._subplots.AxesSubplot at 0x7fa30bcab410>],
 [<matplotlib.axes._subplots.AxesSubplot at 0x7fa30bd5c690>,
  <matplotlib.axes._subplots.AxesSubplot at 0x7fa30bd6eb50>],
 [<matplotlib.axes._subplots.AxesSubplot at 0x7fa30c27ac90>,
  <matplotlib.axes._subplots.AxesSubplot at 0x7fa30bba7750>],
 [<matplotlib.axes._subplots.AxesSubplot at 0x7fa30bbca150>,
  <matplotlib.axes._subplots.AxesSubplot at 0x7fa30b994590>],
 [<matplotlib.axes._subplots.AxesSubplot at 0x7fa30bcd24d0>,
  <matplotlib.axes._subplots.AxesSubplot at 0x7fa30b8c0550>],
 [<matplotlib.axes._subplots.AxesSubplot at 0x7fa30b835c10>,
  <matplotlib.axes._subplots.AxesSubplot at 0x7fa30b835890>],
 [<matplotlib.axes._subplots.AxesSubplot at 0x7fa30b6e4690>,
  <matplotlib.axes._subplots.AxesSubplot at 0x7fa30b60d6d0>],
 [<matplotlib.axes._subplots.AxesSubplot at 0x7fa30c275810>,
  <matplotlib.axes._subplots.AxesSubplot at 0x7fa30c283a10>],
 [<matplotlib.axes._subplots.AxesSubplot at 0x7fa30f1c5a50>,
  <matplotlib.axes._subplots.AxesSubplot at 0x7fa30c679f50>]]
assumptions_results = lifelines.statistics.proportional_hazard_test(
    cph, df_train.loc[:, features_keep], time_transform='rank'
)
assumptions_results.print_summary()
time_transform rank
null_distribution chi squared
degrees_of_freedom 1
model <lifelines.CoxPHFitter: fitted with 4930 total...
test_name proportional_hazard_test
test_statistic p -log2(p)
Contract_One year 44.75 <0.005 35.38
Contract_Two year 36.86 <0.005 29.55
Dependents_Yes 0.02 0.89 0.16
DeviceProtection_Yes 1.64 0.20 2.32
MultipleLines_Yes 14.12 <0.005 12.51
OnlineBackup_Yes 0.02 0.88 0.19
OnlineSecurity_Yes 4.69 0.03 5.04
PaperlessBilling_Yes 0.48 0.49 1.04
Partner_Yes 0.46 0.50 1.00
PaymentMethod_Credit card (automatic) 4.97 0.03 5.27
PaymentMethod_Electronic check 2.27 0.13 2.92
PaymentMethod_Mailed check 0.52 0.47 1.09
SeniorCitizen 4.85 0.03 5.18
TechSupport_Yes 7.02 0.01 6.95
TotalCharges 385.43 <0.005 282.65
gender_Male 0.47 0.49 1.02